ALBERT

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Fabio Punzo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We investigate uniqueness of solutions to the initial value problem for degenerate parabolic equations, posed in bounded domains, where no boundary conditions are prescribed. In order to obtain uniqueness, we need that the solutions satisfy certain integral growth conditions, which are crucially related to the degeneracy of the operator near the boundary. In particular, such solutions can be unbounded near the boundary. Our hypothesis on the behavior of the operator at the boundary is optimal; in fact, we show that if it fails, then nonuniqueness of solutions prevails.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yuhui Chen, Wei Luo, Zheng-an Yao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we mainly investigate the Cauchy problem for the periodic Phan-Thein-Tanner (PTT) model. This model is derived from network theory for the polymeric fluid. We prove that the strong solutions of PTT model will blow up in finite time if the trace of initial stress tensor 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mi mathvariant="normal"〉tr〈/mi〉〈/mrow〉〈mspace width="0.2em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is negative. It is thus very different from the other viscoelastic model. On the other hand, we obtain the global existence result with small initial data when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"〉〈mrow〉〈mi mathvariant="normal"〉tr〈/mi〉〈/mrow〉〈mspace width="0.2em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≥〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Moreover, we study about the large time behavior.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hui Li, Wei Wang, Zhifei Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove the local well-posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition, which ensures that the evolution equation describing the free boundary is strictly hyperbolic. Our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the Rayleigh-Taylor instability.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 1 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Xianbo Sun, Pei Yu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper concerns the exact bound on the number of zeros of Abelian integrals associated with two hyper-elliptic Hamiltonian systems of degree 4. The upper and lower bounds for the two systems have been obtained in several previous works, however the sharp bounds are still unknown. In this paper, we provide a proof to show that the exact bound is 3 for both systems. The basic idea of our method is to bound the parameter space in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 to obtain two parameter sets, which might yield maximal 4 zeros of Abelian integrals corresponding to the two sets. Further, we show that the existence of 4 zeros on the two sets is not possible, and thus the sharp bound is 3.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Bixiang Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 267, Issue 9〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: Available online 9 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Kelei Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove a Liouville type theorem and a partial regularity result for stable and finite Morse index solutions of Toda system. The main tools are integral estimates, a monotonicity formula and blowing down analysis. For such systems, they may be split into two sub-systems during the blowing down process. Some 〈em〉ε〈/em〉-regularity theorems are developed to deal with this problem.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hui Liu, Hongjun Gao〈/p〉

Description: 〈p〉Publication date: Available online 12 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Sandra Cerrai, Alessandra Lunardi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove Schauder type estimates for stationary and evolution equations driven by the classical Ornstein-Uhlenbeck operator in a separable Banach space, endowed with a centered Gaussian measure.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Kaname Matsue〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates. Similarities for characterizing finite-time extinction and asymptotic behavior of compacton traveling waves to blow-up solutions are also shown.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Janusz Mierczyński, Lei Niu, Alfonso Ruiz-Herrera〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 9〈/p〉 〈p〉Author(s): Liang Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉g〈/mi〉〈mo〉,〈/mo〉〈mi〉d〈/mi〉〈mi〉μ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be 〈em〉n〈/em〉-dimensional noncompact metric measure space which satisfies 〈em〉Poincaré〈/em〉 inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mo〉△〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈mi〉c〈/mi〉〈msup〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mo〉≤〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉m〈/mi〉〈mo〉〉〈/mo〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉〈〈/mo〉〈mi〉p〈/mi〉〈mo〉〈〈/mo〉〈mfrac〉〈mrow〉〈mi〉m〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈msqrt〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉m〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉m〈/mi〉〈mo〉+〈/mo〉〈mn〉3〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msqrt〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mi〉σ〈/mi〉〈mo〉≤〈/mo〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/math〉 are real constants.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 9〈/p〉 〈p〉Author(s): Annamaria Canino, Francesco Esposito, Berardino Sciunzi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider positive solutions to semilinear elliptic problems with singular nonlinearity. We provide a Höpf type boundary lemma via a suitable scaling argument that allows to deal with the lack of regularity of the solutions up to the boundary.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 9〈/p〉 〈p〉Author(s): Anca-Voichita Matioc, Bogdan-Vasile Matioc〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting these aspects, we establish the local well-posedness of the problem for arbitrary initial data in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉S〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉3〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, determine a new criterion for the global existence of solutions, and uncover a parabolic smoothing property. Besides, we prove that the zero steady-state solution is exponentially stable.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 9〈/p〉 〈p〉Author(s): Bin Xie〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The hypercontractive property of the Markov semigroup associated with the reflected stochastic partial differential equation driven by the additive space–time white noise is mainly investigated. The main tool for its proof is the general criterion presented recently by F.-Y. Wang [29]. In particular, we obtain the hyperboundedness and the compactness of the Markov semigroup, the exponential convergences of the entropy, the exponential convergences of the Markov semigroup to its unique invariant measure in both 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 and the total variation norm.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Wanrong Yang, Quansen Jiu, Jiahong Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δ〈em〉u〈/em〉 and the magnetic diffusion Δ〈em〉b〈/em〉 by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈mi〉b〈/mi〉〈/math〉, respectively, the resulting equations with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉≥〈/mo〉〈mfrac〉〈mrow〉〈mn〉5〈/mn〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉+〈/mo〉〈mi〉β〈/mi〉〈mo〉≥〈/mo〉〈mfrac〉〈mrow〉〈mn〉5〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉 then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉5〈/mn〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mi〉b〈/mi〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msubsup〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mo〉+〈/mo〉〈msubsup〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Yongki Lee〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We identify sub-thresholds for finite time shock formation in a class of non-local conservation law with concavity changing flux. From a class of non-local conservation laws, the Riccati-type ODE system that governs a solution's gradient is obtained. The changes in concavity of the flux function correspond to the sign changes in the leading coefficient functions of the ODE system. We identify the blow up condition of this structurally generalized Riccati-type ODE. The method is illustrated via the traffic flow models with nonlocal-concave-convex flux. The techniques and ideas developed in this paper is applicable to a large class of non-local conservation laws.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): A.A. Kashchenko〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, nonlocal dynamics of a system of two differential equations with a compactly supported nonlinearity and delay is studied. For some set of initial conditions asymptotics of solutions of considered system is constructed. By this asymptotics we build a special mapping. Dynamics of this mapping describes dynamics of initial system in general: it is proved that stable cycles of this mapping correspond to exponentially orbitally stable relaxation periodic solutions of initial system of delay differential equations. It is shown that amplitude, period of solutions of initial system, and number of coexisting stable solutions depend crucially on coupling parameter. Algorithm for constructing many coexisting stable solutions is described.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Barbara Niethammer, Alessia Nota, Sebastian Throm, Juan J.L. Velázquez〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes 〈em〉v〈/em〉 of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉σ〈/mi〉〈/mrow〉〈/msup〉〈/math〉. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉σ〈/mi〉〈mo〉〉〈/mo〉〈mn〉3〈/mn〉〈/math〉, but the model can be expected to be valid, on heuristic grounds, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉σ〈/mi〉〈mo〉〉〈/mo〉〈mfrac〉〈mrow〉〈mn〉5〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent 〈em〉σ〈/em〉. Here we show that for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mn〉5〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉〈〈/mo〉〈mi〉σ〈/mi〉〈mo〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉 the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Anders Björn, Jana Björn〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉X〈/em〉 be a noncomplete metric measure space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mover accent="true"〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉ˆ〈/mo〉〈/mrow〉〈/mover〉〈/math〉 of 〈em〉X〈/em〉 and use them to obtain several results on 〈em〉X〈/em〉 itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to 〈em〉p〈/em〉-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Kui Li, Zhitao Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉, we prove this conjecture is true for space dimension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉=〈/mo〉〈mn〉3〈/mn〉〈/math〉; which also implies the single elliptic equation has no positive classical solutions in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Hong Tian, Shenzhou Zheng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove a global Lorentz estimate for the variable power of the gradients of weak solution to parabolic obstacle problems with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-growth over a bounded nonsmooth domain. It is mainly assumed that the variable exponents 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 satisfy a strong type log-Hölder continuity, the associated nonlinearities are merely measurable in the time variable and have small BMO semi-norms in the spatial variables, while the underlying domain is quasiconvex.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Chiun-Chang Lee〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For the structure of the thin electrical double layer (EDL) and the property related to the EDL capacitance, we analyze boundary layer solutions (corresponding to the electrostatic potential) of a non-local elliptic equation which is a steady-state Poisson–Nernst–Planck equation with a singular perturbation parameter related to the small Debye screening length. Theoretically, the boundary layer solutions describe that those ions exactly approach neutrality in the bulk, and the extra charges are accumulated near the charged surface. Hence, the non-neutral phenomenon merely occurs near the charged surface. To investigate such phenomena, we develop new analysis techniques to investigate thin boundary layer structures. A series of fine estimates combining the Pohožaev's identity, the inverse Hölder type estimates and some technical comparison arguments are developed in arbitrary bounded domains. Moreover, we focus on the physical domain being a ball with the simplest geometry and gain a clear picture on the effect of the curvature on the boundary layer solutions. The content involves three contributions. The first one focuses mainly on the boundary concentration phenomena. We show that the net charge density behaves exactly as Dirac delta measures concentrated at boundary points. The second one is devoted to pointwise descriptions with curvature effects for the thin boundary layer. An interesting outcome shows that the significant curvature effect merely occurs in the part of the boundary layer close to the boundary, and this part is extremely thinner than the whole boundary layer. The third contribution gives a connection to the EDL capacitance. We provide a theoretical way to support that the EDL has higher capacitance in a quite thin region near the charged surface, not in the whole EDL. In particular, for the cylindrical electrode, our result has a same analogous measurement as the specific capacitance of the well-known Helmholtz double layer.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): Junmin Yang, Pei Yu, Maoan Han〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order 〈em〉m〈/em〉 in a near-Hamiltonian system. For any positive integer 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉m〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo〉=〈/mo〉〈mn〉2〈/mn〉〈/math〉, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 January 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 1〈/p〉 〈p〉Author(s): H. Bueno, O.H. Miyagaki, G.A. Pereira〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉With appropriate hypotheses on the nonlinearity 〈em〉f〈/em〉, we prove the existence of a ground state solution 〈em〉u〈/em〉 for the problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈mi〉W〈/mi〉〈mo〉⁎〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mtext〉in 〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉σ〈/mi〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉, 〈em〉V〈/em〉 is a bounded continuous potential and 〈em〉F〈/em〉 the primitive of 〈em〉f〈/em〉. We also show results about the regularity of any solution of this problem.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): Gianni Dal Maso, Rodica Toader〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous definition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Bas Lemmens, Onno van Gaans, Hent van Imhoff〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we prove a recent conjecture by M. Hirsch, which says that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉f〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a discrete time monotone dynamical system, with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉f〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/math〉 a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of 〈em〉f〈/em〉 are dense in Ω, then 〈em〉f〈/em〉 is periodic.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Junfeng He, Yaping Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate Fisher type equation in an unbounded cylinder. Firstly, by applying the moving plane argument and the generalized center manifold theorem, we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non-exponential decay of traveling fronts with non-critical speeds, especially for the 〈em〉p〈/em〉-degree Fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non-critical speeds. Secondly, by applying the spectral analysis and sub-super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the Lyapunov stability of traveling fronts with non-critical speeds in some polynomially weighted spaces. Finally, by combining the sub-super solution method with the nonlinear stability results, we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data, which are proved to be determined by the spatial decay of the initial data at one end.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): C. Campana, P.L. Dattori da Silva, A. Meziani〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Enzo Vitillaro〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉The aim of this paper is to study the problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mrow〉〈mo〉{〈/mo〉〈mtable〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="2em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉in 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉on 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉Q〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="2em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉on 〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mtext〉in 〈/mtext〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mtext〉,〈/mtext〉〈/mrow〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where Ω is a bounded open subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 boundary (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉), 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Γ〈/mi〉〈mo〉=〈/mo〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is relatively open on Γ, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉Γ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 denotes the Laplace–Beltrami operator on Γ, 〈em〉ν〈/em〉 is the outward normal to Ω, and the terms 〈em〉P〈/em〉 and 〈em〉Q〈/em〉 represent nonlinear damping terms, while 〈em〉f〈/em〉 and 〈em〉g〈/em〉 are nonlinear perturbations.〈/p〉 〈p〉In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): K. Dareiotis, M. Gerencsér, B. Gess〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 for all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, and Hölder continuous diffusion nonlinearity with exponent 1/2.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 5 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 8〈/p〉 〈p〉Author(s): Carlo Orrieri, Luca Scarpa〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using 〈em〉a priori〈/em〉 estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Yujin Guo, Yong Luo, Qi Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-constraint minimizers of the mass critical Hartree energy functional with a trapping potential 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉V〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in a bounded domain Ω of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/math〉. We prove that minimizers exist if and only if the parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 satisfies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mo〉〈〈/mo〉〈msup〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈msubsup〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈mi〉Q〈/mi〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉Q〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 is the unique positive solution of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉(〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉y〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉)〈/mo〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/math〉. By investigating new analytic methods, the refined limit behavior of minimizers as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si162.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mo stretchy="false"〉↗〈/mo〉〈msup〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉 is analyzed for both cases where all the mass concentrates either at an inner point 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 of Ω or near the boundary of Ω, depending on whether 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉V〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 attains its flattest global minimum at an inner point 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 of Ω or not. As a byproduct, we also establish two Gagliardo–Nirenberg type inequalities which are of independent interest.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Christos Sourdis〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that the kernel of the linearization of the blow-up problem at the regular part of the interface that separates segregated BECs is one-dimensional, generated by translations in the normal direction to the interface. This useful non-degeneracy property was previously known only in one and two dimensions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): A. Aghajani, C. Cowan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work we obtain positive singular solutions of〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mrow〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable columnspacing="0em"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mspace width="2em"〉〈/mspace〉〈mtext〉 in 〈/mtext〉〈mi〉y〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mtext〉 on 〈/mtext〉〈mi〉y〈/mi〉〈mo〉∈〈/mo〉〈mo〉∂〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is a sufficiently small 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 perturbation of the cone 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉:〈/mo〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉=〈/mo〉〈mi〉r〈/mi〉〈mi〉θ〈/mi〉〈mo〉,〈/mo〉〈mi〉r〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉θ〈/mi〉〈mo〉∈〈/mo〉〈mi〉S〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉S〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 has a smooth nonempty boundary and where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉 satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): G. Barbatis, P. Branikas〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉 coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this work is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): David M. Ambrose, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this article we consider weak solutions of the Euler-〈em〉α〈/em〉 equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉O〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mi mathvariant="normal"〉log〈/mi〉〈mo〉⁡〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and 〈em〉p〈/em〉-th power integrable, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo〉〉〈/mo〉〈mn〉2〈/mn〉〈/math〉, due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Alexander Sakhnovich〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 8〈/p〉 〈p〉Author(s): Siyu Liu, Haomin Huang, Mingxin Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Recently, the authors of [22] studied a diffusive prey–predator model with two different free boundaries. They first obtained the existence, uniqueness, regularity, uniform estimates and long time behaviors of global solution, and then established the conditions for spreading and vanishing. Especially, when spreading occurs, they provided accurate limits of two species as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mo〉+〈/mo〉〈mo〉∞〈/mo〉〈/math〉, and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries. Motivated by the paper [22], in this paper we discuss the diffusive competition model with two different free boundaries, which had been investigated by [7], [11], [15], [21]. The main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs. Furthermore, how the solution approaches the semi-wave when spreading happens is also described.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Yongxia Guo, Guangsheng Wei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we provide the sharp conditions of the uniqueness for inverse nodal Sturm–Liouville problems defined on interval 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉 with separated boundary conditions. We prove that the potential and boundary parameters can be uniquely determined by a dense nodal subset contained on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉[〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉]〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⊂〈/mo〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.gif" overflow="scroll"〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉2〈/mn〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 through two cases of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si36.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, where in the latter case the nodal subset also need to be paired. Note that, the dense nodal subset was required to be twin for both cases in the previous works.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 8〈/p〉 〈p〉Author(s): Silvia Cingolani, Giuseppina Vannella〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider the quasilinear critical problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable columnspacing="0em"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉on〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where Ω is a regular bounded domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mn〉1〈/mn〉〈mo〉〈〈/mo〉〈mi〉p〈/mi〉〈mo〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo〉≤〈/mo〉〈mi〉q〈/mi〉〈mo〉〈〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈mi〉N〈/mi〉〈mi〉p〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mi〉p〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si214.gif" overflow="scroll"〉〈mi〉λ〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 is a parameter. In spite of the lack of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si494.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 regularity of the energy functional associated to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 such that, for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mi〉λ〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, either 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 has 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 distinct solutions or there exists a sequence of quasilinear problems approximating 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, each of them having at least 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 distinct solutions. These results complete those obtained in [23] for the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Huan Liu, Xianguo Geng, Bo Xue〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The initial value problem for the Sasa–Satsuma equation is transformed to a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mn〉3〈/mn〉〈mo〉×〈/mo〉〈mn〉3〈/mn〉〈/math〉 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump matrix and a decomposition of the vector-valued function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.gif" overflow="scroll"〉〈mi〉ρ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Josiney A. Souza, Luana H. Takamoto〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Juhi Jang, Tetu Makino〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider stationary axisymmetric solutions of the Euler–Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mn〉6〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉5〈/mn〉〈mo〉〈〈/mo〉〈mi〉γ〈/mi〉〈mo〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉 and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implicit function theorem. Oblateness of star surface is shown and also relationship between the central density and the total mass is given.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Chenchen Mou, Andrzej Święch〈/p〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): Pu-Zhao Kow, Ching-Lung Lin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we consider the asymptotic behavior of an incompressible fluid around a bounded obstacle. By adapting the Schauder's estimate for stationary Navier–Stokes equation to improve the regularity, the problem is solved by using appropriate Carleman estimates. It should be noted that the minimal decaying rate for a general scalar equation is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="normal"〉exp〈/mi〉〈mo〉⁡〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉C〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. However, the structure of the Navier–Stokes is special. Under the assumption for any nontrivial solution to be uniform bounded which is weaker than those in [10], we got the minimal decaying rate is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="normal"〉exp〈/mi〉〈mo〉⁡〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉C〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 which is better than the results in general scalar cases.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Mythily Ramaswamy, Jean-Pierre Raymond, Arnab Roy〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): Jonathan Jaquette〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove that Wright's equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 has a unique slowly oscillating periodic solution for parameter values 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mfrac〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mn〉1.9〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉, up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉〉〈/mo〉〈mfrac〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): Shengliang Pan, Yunlong Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Michael Ruzhansky, Niyaz Tokmagambetov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and on the Heisenberg group, such as powers of the Laplacian or the sub-Laplacian. In addition, we establish a new family of Gagliardo–Nirenberg inequalities on a graded Lie groups that play a crucial role in the proof, but which are also of interest on their own: if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉G〈/mi〉〈/math〉 is a graded Lie group of homogeneous dimension 〈em〉Q〈/em〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si247.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mn〉1〈/mn〉〈mo〉〈〈/mo〉〈mi〉r〈/mi〉〈mo〉〈〈/mo〉〈mfrac〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mn〉1〈/mn〉〈mo〉≤〈/mo〉〈mi〉p〈/mi〉〈mo〉≤〈/mo〉〈mi〉q〈/mi〉〈mo〉≤〈/mo〉〈mfrac〉〈mrow〉〈mi〉r〈/mi〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mi〉Q〈/mi〉〈mo〉−〈/mo〉〈mi〉a〈/mi〉〈mi〉r〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉, then we have the following Gagliardo–Nirenberg type inequality〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈mo〉≲〈/mo〉〈msup〉〈mrow〉〈msub〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈msubsup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉˙〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈msubsup〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉−〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉〈/span〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/mfrac〉〈mo stretchy="false"〉)〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mfrac〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉+〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfrac〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉 provided that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si256.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉+〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉≠〈/mo〉〈mn〉0〈/mn〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉˙〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉 is the homogeneous Sobolev space of order 〈em〉a〈/em〉 over 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msup〉〈/math〉. If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉+〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉, we have 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉p〈/mi〉〈mo〉=〈/mo〉〈mi〉q〈/mi〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉r〈/mi〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mi〉Q〈/mi〉〈mo〉−〈/mo〉〈mi〉a〈/mi〉〈mi〉r〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉, and then the above inequality holds for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.gif" overflow="scroll"〉〈mn〉0〈/mn〉〈mo〉≤〈/mo〉〈mi〉s〈/mi〉〈mo〉≤〈/mo〉〈mn〉1〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Jin Takahashi, Hikaru Yamamoto〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the heat equation with a superlinear absorption term 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and study the existence of nonnegative solutions with an 〈em〉m〈/em〉-dimensional time-dependent singular set, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉m〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/math〉. We prove that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mn〉1〈/mn〉〈mo〉〈〈/mo〉〈mi〉p〈/mi〉〈mo〉〈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉m〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉m〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, then there are two types of singular solutions. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 6〈/p〉 〈p〉Author(s): Yajing Li, Yejuan Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We first prove the existence, uniqueness and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative:〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mmultiscripts〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mprescripts〉〈/mprescripts〉〈none〉〈/none〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈/mmultiscripts〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉A〈/mi〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mfrac〉〈mrow〉〈mi〉d〈/mi〉〈mi mathvariant="double-struck"〉W〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉〈〈/mo〉〈mi〉α〈/mi〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mo〉.〈/mo〉〈/math〉〈/span〉 Then, we investigate the asymptotic behavior of mild solutions to fractional stochastic delay evolution equations of the form〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mmultiscripts〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mprescripts〉〈/mprescripts〉〈none〉〈/none〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈/mmultiscripts〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉A〈/mi〉〈mi〉y〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈msubsup〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈mo stretchy="false"〉[〈/mo〉〈msubsup〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉g〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉]〈/mo〉〈mfrac〉〈mrow〉〈mi〉d〈/mi〉〈mi mathvariant="double-struck"〉W〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mrow〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉α〈/mi〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mo〉.〈/mo〉〈/math〉〈/span〉 In particular, the existence of a global forward attracting set in the mean-square topology is established. A general theorem on the existence of mild solutions is obtained by using 〈em〉α〈/em〉-order fractional resolvent operator theory and the Schauder fixed point theorem.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Zhong Tan, Wenpei Wu, Jianfeng Zhou〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mo〉˙〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉≤〈/mo〉〈mi〉s〈/mi〉〈mo〉〈〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 or homogeneous Besov norms 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mo〉˙〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉s〈/mi〉〈mo〉≤〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉≤〈/mo〉〈mi〉p〈/mi〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 type of the decay rates without requiring that the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with large initial data.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Ki-Ahm Lee, Jinwan Park〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study a priori 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 estimate up to the boundary for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉F〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator. The conditions are variations of conditions for the zero set of the operator in [7].〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Esther S. Daus, Laurent Desvillettes, Helge Dietert〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada–Kawasaki–Teramoto (SKT) type (cf. [1]) satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Shi-Liang Wu, Guang-Sheng Chen, Cheng-Hsiung Hsu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the entire solutions of a nonlocal dispersal epidemic model which arises from the spread of fecally–orally transmitted diseases. Under bistable assumptions, it is well-known that this model has three different types of traveling wave fronts. The annihilating-front and merging-front entire solutions originating from 〈em〉two〈/em〉 fronts of the system have also been constructed in [38]. We first prove the uniqueness, Liapunov stability and continuous dependence on shift parameters of annihilating-front entire solutions obtained in [38]. A positive time-derivative estimate for such entire solution is also obtained. Then, we establish the existence of two different types of entire solutions merging 〈em〉three different fronts〈/em〉. Furthermore, we show that these entire solutions are global Lipschitz continuous with respect to the spatial variable 〈em〉x〈/em〉. To the best of our knowledge, it is the first time that the entire solutions originating from three fronts of diffusion systems have been constructed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Elisa Affili, Enrico Valdinoci〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.〈/p〉 〈p〉Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 5 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 11〈/p〉 〈p〉Author(s): Hebai Chen, Sen Duan, Yilei Tang, Jianhua Xie〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The global dynamics of a mechanical system with dry friction is completely analyzed. This mechanical system is a class of discontinuous and transcendental piecewise smooth differential systems. Moreover, it can exhibit rich and complex dynamical phenomena, such as Hopf bifurcation, grazing bifurcation, grazing-sliding bifurcation and bifurcations of limit cycles. Finally, all global phase portraits of the system are presented on the Poincaré disc.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉 〈p〉Author(s): Xijun Hu, Lei Liu, Li Wu, Hao Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉It is natural to consider continuous dependence of the 〈em〉n〈/em〉-th eigenvalue on 〈em〉d〈/em〉-dimensional (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉d〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉) Sturm–Liouville problems after the results on 1-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the 〈em〉n〈/em〉-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the 〈em〉n〈/em〉-th eigenvalue. This renders a precise description of the jump phenomena of the 〈em〉n〈/em〉-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mn〉2〈/mn〉〈mi〉d〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/math〉 layers and show that the 〈em〉n〈/em〉-th eigenvalue is continuously dependent on Sturm–Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 8〈/p〉 〈p〉Author(s): Y.Sh. Ilyasov, N.F. Valeev〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We establish a relationship between an inverse optimization spectral problem for the N-dimensional Schrödinger equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉ϕ〈/mi〉〈mo〉+〈/mo〉〈mi〉q〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉ϕ〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉ϕ〈/mi〉〈/math〉 and a solution of the nonlinear boundary value problem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉q〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉γ〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mi〉u〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Zhipeng Qiu, Michael Y. Li, Zhongwei Shen〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we derive and analyze a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. We establish the well-posedness and dissipativity of the associated nonlinear semigroup. We establish an equivalent principal spectral condition between the linearized operator and the next-generation operator and show that the basic reproduction number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a sharp threshold: if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉, the disease-free equilibrium is globally asymptotically stable, and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉, the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. The proof of global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on networks developed in [24].〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Aníbal Rodríguez-Bernal, Alejandro Vidal-López〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we derive the limiting PDES for parabolic problems where localized large diffusion touches the boundary of the domain. Hence the limit problem reflects the interaction of large diffusion and boundary conditions. The limit problem is a PDE coupled with a system of ODEs through some nonlocal terms. We study the well posedness and dissipativity of the (nonstandard) limit problem as well as the continuity of the dynamics as diffusion becomes large.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Myoungjean Bae, Hyangdong Park〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove the existence of a subsonic axisymmetric weak solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉ρ〈/mi〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi mathvariant="bold"〉e〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi mathvariant="bold"〉e〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi mathvariant="bold"〉e〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 to steady Euler system in a three-dimensional infinitely long cylinder 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉N〈/mi〉〈/math〉 when prescribing the values of the entropy 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈mrow〉〈mi〉γ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and angular momentum density 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mo〉=〈/mo〉〈mi〉r〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 at the entrance by piecewise 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 functions with a discontinuity on a curve on the entrance of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉N〈/mi〉〈/math〉. Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.gif" overflow="scroll"〉〈mi〉r〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a solution and analyze the asymptotic behavior of the solution at far field.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Xian-Gao Liu, Jianzhong Min, Xiaotao Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this article, we study the simplified system of the original Ericksen–Leslie equations for the flow of liquid crystals. It is shown that the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉 solutions of the Cauchy problem for the three-dimensional incompressible liquid crystal system are smooth.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Peter Hästö, Jihoon Ok〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We establish the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉φ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉-solvability of the linear elliptic equations in non-divergence form under a suitable, essentially minimal, condition of the generalized Orlicz function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉φ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉φ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, by deriving Calderón–Zygmund type estimates. The class of generalized Orlicz spaces we consider here contains as special cases classical Lebesgue and Orlicz spaces, as well as non-standard growth cases like variable exponent and double phase growth.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Yilei Tang, Weinian Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Recently attentions were paid to versal unfolding of degenerate equilibria within a restricted family of vector fields for a practical sense. The family of Liénard systems is such a family of significant physical sense but only within the family of even Liénard systems there were found results for a nilpotent degenerate equilibrium. In this paper we discuss versal unfolding of a nilpotent Liénard equilibrium within the family of odd Liénard systems. As the well-known Bogdanov-Takens normal form is not available for the odevity, we prove that the degeneracy within the family is of codimension 2 and find its versal unfolding. We further discuss the unfolding system, displaying all its bifurcations such as pitchfork bifurcation, center-saddle bifurcation, center bifurcation and homoclinic (heteroclinic) loop bifurcation, which show how a single homoclinic loop, a twin homoclinic loop and a heteroclinic loop arise from a center or a (degenerate) saddle.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Huijun He, Zhaoyang Yin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo〉=〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉−〈/mo〉〈msubsup〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. Firstly, we obtain the Gevrey regularity and analyticity for a short time. Secondly, we show the continuity of the data-to-solution map. Finally, we prove the global Gevrey regularity and analyticity in time.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 26 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Zdzisław Brzeźniak, Erika Hausenblas, Liang Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider 1D Landau-Lifshitz-Gilbert equations with an external force. We first prove the existence and uniqueness of the strong solution, and then give a definition of the quasipotential and prove that the quasipotential is equal to the potential energy of the system.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Zhuan Ye〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we are concerned with the Cauchy problem of the multi-dimensional inhomogeneous incompressible magnetohydrodynamic equations with fractional dissipation. We prove the unique global strong solution with vacuum to the magnetohydrodynamic equations over the whole space. In addition, the large time decay rates of the corresponding strong solution are also obtained. Note that our result is proved without any smallness on the initial data. Moreover, the initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. To our best knowledge, this is the first result on the inhomogeneous fractional magnetohydrodynamic equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Hongwei Gao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton–Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Boling Guo, Binqiang Xie, Lan Zeng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the dynamics of an Boussinesq approximation Bénard convection fluid evolving in a three-dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface. The domain is horizontally periodic and the effect of the surface tension is on the free surface. By developing a priori estimates for the model, we prove the exponential decay of solutions in the framework of high regularity.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 10〈/p〉 〈p〉Author(s): Jaewook Ahn〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A fully parabolic chemotaxis system〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈mi〉u〈/mi〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉v〈/mi〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mspace width="2em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo〉−〈/mo〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈/math〉〈/span〉 in a smooth bounded domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉 with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function 〈em〉χ〈/em〉 satisfies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉μ〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉a〈/mi〉〈mo〉+〈/mo〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/math〉, for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mo〉≥〈/mo〉〈mn〉0〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉k〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈/math〉. It is shown that a novel type of weight function can be applied to a weighted energy estimate for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si56.gif" overflow="scroll"〉〈mi〉k〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. Consequently, the range of 〈em〉μ〈/em〉 for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉k〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈/math〉 under a smallness assumption on 〈em〉μ〈/em〉. In particular, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉μ〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉v〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉〈〈/mo〉〈mn〉8〈/mn〉〈/math〉, it is shown that the spatially homogeneous steady state is a global attractor whenever 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉μ〈/mi〉〈mo〉≤〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉2〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 7 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Dietmar Hömberg, Shuai Lu, Masahiro Yamamoto〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 3 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): In-Jee Jeong〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 3 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Lassi Paunonen, David Seifert〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the one-dimensional wave equation with periodic damping.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 3 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Kun-Peng Jin, Jin Liang, Ti-Jun Xiao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function 〈em〉g〈/em〉, without the usual assumption of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉 controlled by 〈em〉g〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 3 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Shaobo Gan, Yi Shi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In 1994, I. Kan constructed a smooth map on the annulus admitting two physical measures, whose basins are intermingled. In this paper, we prove that Kan's map is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 robustly topologically mixing.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations〈/p〉 〈p〉Author(s): Shimin Li, Jaume Llibre〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): K. da S. Andrade, M.R. Jeffrey, R.M. Martins, M.A. Teixeira〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper presents a version of Dulac's Problem for piecewise analytic vector fields that states conditions for the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): Riikka Korte, Pekka Lehtelä, Stefan Sturm〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We deal with the obstacle problem for the porous medium equation in the slow diffusion regime 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): Mehdi Pourbarat〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): Benjamin Kennedy〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the real-valued differential equation〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉−〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉〈/span〉 with state-dependent delay, where 〈em〉f〈/em〉 is strictly monotonic in its second argument. We describe a class of such equations for which a version of the Poincaré–Bendixson theorem holds.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): Razvan C. Fetecau, Hui Huang, Weiran Sun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 5〈/p〉 〈p〉Author(s): Zbigniew Galias, Warwick Tucker〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 5〈/p〉 〈p〉Author(s): Adam Larios, Yuan Pei, Leo Rebholz〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.〈/p〉〈/div〉

Description: 〈p〉Publication date: 5 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 4〈/p〉 〈p〉Author(s): Xing Liang, Lei Zhang, Xiao-Qiang Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.〈/p〉〈/div〉