ALBERT

Description: 〈p〉Publication date: Available online 25 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Mohamed Karim Hamdani, Dušan D. Repovš〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the following 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-curl systems:〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si103.svg"〉〈mrow〉〈mrow〉〈mo〉{〈/mo〉〈mtable〉〈mtr〉〈mtd columnalign="left"〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈mi〉μ〈/mi〉〈mi〉g〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mtext〉 in 〈/mtext〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="bold"〉n〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi mathvariant="bold"〉u〈/mi〉〈mo〉⋅〈/mo〉〈mi mathvariant="bold"〉n〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mtext〉 on 〈/mtext〉〈mo〉∂〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 is a bounded simply connected domain with a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈/math〉 boundary, denoted by ∂Ω, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.svg"〉〈mi〉p〈/mi〉〈mo〉:〈/mo〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉→〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous function, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉a〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.svg"〉〈mi〉f〈/mi〉〈mo〉,〈/mo〉〈mi〉g〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉→〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 are Carathéodory functions, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉μ〈/mi〉〈/math〉 are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang, Wang and Zhang (J. Math. Anal. Appl., 2016), Bahrouni and Repovš (Complex Var. Elliptic Equ., 2018), and Bin and Fang (Mediterr. J. Math., 2019).〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Deepak Kumar, K. Sreenadh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem:〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable〉〈mtr〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉β〈/mi〉〈msubsup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈munder〉〈mo movablelimits="false"〉∫〈/mo〉〈mrow〉〈mi mathvariant="normal"〉Ω〈/mi〉〈/mrow〉〈/munder〉〈mfrac〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉y〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mspace width="0.25em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉∖〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is a bounded domain with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 boundary ∂Ω, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.svg"〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉2〈/mn〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉r〈/mi〉〈mo〉≤〈/mo〉〈msubsup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msubsup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/mfrac〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si60.svg"〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉a〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉q〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msubsup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈mo〉:〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈msub〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/mfrac〉〈/math〉, is a sign changing function. We prove that each nonnegative weak solution of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Mingwen Fei, Dongjuan Niu, Xiaoming Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We investigate the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. In particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. Of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): A. Ambrazevičius, V. Skakauskas〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The purpose of this paper is to investigate the existence and uniqueness of classical solutions to a model of surface reactions between two polyatomic reactants. The model is described by a coupled system of four quasilinear parabolic equations where two of them are determined in the domain and the other two are considered on a part of its surface. The elliptic operators of the parabolic equations determined on the surface are allowed to be degenerate in the sense that the density-dependent diffusion coefficients 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 may have the property 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si178.svg"〉〈mi〉i〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 23 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Nikolaos S. Papageorgiou, Dušan D. Repovš, Calogero Vetro〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a 〈em〉p〈/em〉-Laplacian and of a weighted 〈em〉q〈/em〉-Laplacian (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈/math〉) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, the equation has at least two positive solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Xiaofen Gao, Chengbin Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the long time behavior of the solution of nonlinear Schrödinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉i〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/math〉〈/span〉 when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mn〉2〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mfrac〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈/math〉. This work extends the result in [12], [15], [16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2] and that the equivalence of Sobolev norms associated to the operator 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈/msub〉〈/math〉 and the operator 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈/math〉 with Aharonov-Bohn potential for radial function.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Enrique Jordá, Alberto Rodríguez-Arenas〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The composition operators defined on little Bloch spaces, Bergman spaces, Hardy spaces or little weighted Bergman spaces of infinite type, when well defined, are shown to be mean ergodic if and only if they are power bounded if and only if the symbol has an interior fixed point. For these operators uniform mean ergodicity is equivalent to quasicompactness in the sense of Yosida and Kakutani.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Paweł Pasteczka〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We will prove that every pair of quasi-arithmetic means satisfying certain smoothness assumption has the supremum and the infimum in this set. More precisely, if 〈em〉f〈/em〉 and 〈em〉g〈/em〉 are 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="script"〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 functions with nowhere vanishing first derivative then there exists a function 〈em〉h〈/em〉 such that: (i) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉h〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈/math〉, (ii) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉h〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈/math〉, and (iii) for every continuous strictly monotone function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉s〈/mi〉〈mo〉:〈/mo〉〈mi〉I〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mspace width="0.25em"〉〈/mspace〉〈mtext mathvariant="normal"〉and〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mspace width="0.25em"〉〈/mspace〉〈mtext mathvariant="normal"〉implies〈/mtext〉〈mspace width="0.25em"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉h〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈/math〉〈/span〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉[〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈/mrow〉〈/msup〉〈/math〉 stands for a quasi-arithmetic mean). Moreover, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mi〉h〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="script"〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo〉≠〈/mo〉〈mn〉0〈/mn〉〈/math〉, and it is a solution of the differential equation〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.svg"〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mo〉″〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mi mathvariant="normal"〉max〈/mi〉〈mo〉⁡〈/mo〉〈mo stretchy="true" maxsize="3.8ex" minsize="3.8ex"〉(〈/mo〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mo〉″〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉″〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo stretchy="true" maxsize="3.8ex" minsize="3.8ex"〉)〈/mo〉〈mo〉.〈/mo〉〈/math〉〈/span〉 We also provide some extension to a finite family of means and dual (reflected) result.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Robert E. Gaunt〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The modified Lommel function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in terms of the ratio 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"〉〈mi〉x〈/mi〉〈msubsup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, the ratio 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and the modified Lommel function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 itself that are given in terms of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉x〈/mi〉〈msubsup〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉ν〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, respectively. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Shaoyu Dai, Yifei Pan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Using Hörmander 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈msup〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉a〈/mi〉〈/math〉 of any order in weighted Hilbert space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si144.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msup〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and prove the existence of a right inverse that is bounded.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 1〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Xinyu Tu, Shuyan Qiu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present study, we consider the chemotaxis system with logistic-type superlinear degradation〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable columnspacing="0em"〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉Δ〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉χ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msubsup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msubsup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉t〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉Δ〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉χ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉∇〈/mi〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉v〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msubsup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msubsup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉t〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mn〉0〈/mn〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo〉−〈/mo〉〈mi〉γ〈/mi〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉t〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mrow〉〈/math〉〈/span〉 under the homogeneous Neumann boundary condition, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉γ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉χ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg"〉〈msub〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉i〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. Consider an arbitrary ball 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈mo〉,〈/mo〉〈mi〉R〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉(〈/mo〉〈mi〉i〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, it is shown that for any parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mover accent="true"〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉ˆ〈/mo〉〈/mrow〉〈/mover〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi mathvariant="normal"〉max〈/mi〉〈mo〉⁡〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 satisfies〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mover accent="true"〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉ˆ〈/mo〉〈/mrow〉〈/mover〉〈mo linebreak="badbreak" linebreakstyle="after"〉〈〈/mo〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable〉〈mtr〉〈mtd columnalign="left"〉〈mfrac〉〈mrow〉〈mn〉7〈/mn〉〈/mrow〉〈mrow〉〈mn〉6〈/mn〉〈/mrow〉〈/mfrac〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mtext〉if〈/mtext〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉3〈/mn〉〈mo〉,〈/mo〉〈mn〉4〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/mfrac〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mtext〉if〈/mtext〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉5〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 there exist nonnegative radially symmetric initial data under suitable conditions such that the corresponding solutions blow up in finite time in the sense that〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈munder〉〈mrow〉〈mi mathvariant="normal"〉lim sup〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈mo stretchy="false"〉↗〈/mo〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈mi〉a〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/munder〉〈mspace width="0.2em"〉〈/mspace〉〈mrow〉〈mo stretchy="true"〉(〈/mo〉〈msub〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⋅〈/mo〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈msub〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⋅〈/mo〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈mo stretchy="true"〉)〈/mo〉〈/mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mo〉∞〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mtext〉for some〈/mtext〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.2em"〉〈/mspace〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈mi〉a〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈mo〉.〈/mo〉〈/math〉〈/span〉 Furthermore, for any smooth bounded domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si114.svg"〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉(〈/mo〉〈mi〉i〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, we prove that the system admits a unique global bounded solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Oscar Jarrín〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The Liouville problem for the stationary Navier-Stokes equations on the whole space is a challenging open problem who has known several recent contributions. We prove here some Liouville type theorems for these equations provided the velocity field belongs to some Lorentz spaces and then in the more general setting of Morrey spaces. Our theorems correspond to an improvement of some recent results on this problem and contain some well-known results as a particular case.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Dan Dai, Mourad E.H. Ismail, Xiang-Sheng Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we derive some properties of a Ramanujan type entire function. A mild generalization of the Garret-Ismail-Stanton 〈em〉m〈/em〉-version of the Rogers-Ramanujan identities is obtained. Moreover, we investigate the zeros of the Ramanujan type entire function, and our results generalize those for the zeros of the Ramanujan function. Finally, an integral equation related to the Ramanujan type entire function is also derived.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Vidhya Krishnasamy, Lakshmi Sankar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the problem〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mo stretchy="true"〉{〈/mo〉〈mtable displaystyle="true" columnspacing="0.025em"〉〈mtr〉〈mtd columnalign="right"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈/mtd〉〈mtd columnalign="left"〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉K〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.20in"〉〈/mspace〉〈mtext〉in 〈/mtext〉〈msubsup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msubsup〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈mi〉u〈/mi〉〈/mtd〉〈mtd columnalign="left"〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="0.78in"〉〈/mspace〉〈mtext〉on 〈/mtext〉〈mo〉∂〈/mo〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mo stretchy="false"〉→〈/mo〉〈mn〉0〈/mn〉〈mspace width="0.76in"〉〈/mspace〉〈mtext〉as 〈/mtext〉〈mrow〉〈mo stretchy="true"〉|〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="true"〉|〈/mo〉〈/mrow〉〈mo stretchy="false"〉→〈/mo〉〈mo〉∞〈/mo〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msubsup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msubsup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mspace width="0.25em"〉〈/mspace〉〈mo stretchy="false"〉|〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mrow〉〈mo stretchy="true"〉|〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="true"〉|〈/mo〉〈/mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈mo〉,〈/mo〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉2〈/mn〉〈/math〉, 〈em〉λ〈/em〉 is a positive parameter, 〈em〉K〈/em〉 belongs to a class of functions which satisfy certain decay assumptions and 〈em〉f〈/em〉 belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter 〈em〉λ〈/em〉. Existence results to similar problems in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Xiao Yan, Yanling Li, Gaihui Guo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the diffusive predator-prey model with toxins subject to Dirichlet boundary conditions. The uniform persistence of positive solution is given under certain conditions. In addition, by the Liapunov-Schmidt method, the existence and stability of the bifurcation solution from a double eigenvalues are investigated. Moreover, by the fixed point index theory and perturbation theory of eigenvalues, the uniqueness, stability and multiplicity of coexistence states are analyzed when some key parameter changes. Finally, some numerical simulations are presented to verify the theoretical conclusions and further to reflect the importance of parameters to the number of coexistence states.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Vladimir I. Bogachev, Ilya I. Malofeev〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always achieved). Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter. As a corollary we obtain the Borel measurability with respect to the parameter for disintegrations of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Duc Quang Si〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉M〈/em〉 be a complete Kähler manifold whose universal covering is biholomorphic to a ball 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉≤〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈/math〉. In this paper, we will show that if two meromorphic mappings 〈em〉f〈/em〉 and 〈em〉g〈/em〉 from 〈em〉M〈/em〉 into 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 have the same inverse images for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉2〈/mn〉〈/math〉 hyperplanes 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msubsup〉〈mrow〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉}〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉 with multiplicities counted to level 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi〉l〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and satisfy the condition 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si48.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉ρ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 then the map 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.svg"〉〈mi〉f〈/mi〉〈mo〉×〈/mo〉〈mi〉g〈/mi〉〈/math〉 is algebraically degenerate over 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si147.svg"〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi〉l〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a positive integer and 〈em〉ρ〈/em〉 is a non-negative number (explicitly estimated). Our result generalizes the previous result of H. Fujimoto for the case of mappings on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈/math〉 to the case of mappings on a complete Kähler manifold as above 〈em〉M〈/em〉 and also improves his result by giving an explicit estimate for the number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi〉l〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Chao Liang, Sheng Qian, Wenxiang Sun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For a measure 〈em〉μ〈/em〉 preserved by a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈msub〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈/math〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉) diffeomorphism 〈em〉f〈/em〉 and a continuous function 〈em〉ϕ〈/em〉 on the manifold, we study the relationship between the exponential growth rate of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈mo〉∑〈/mo〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mi〉ϕ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉 over the orbits of some periodic points and the free energy 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉f〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mo〉∫〈/mo〉〈mi〉ϕ〈/mi〉〈mi〉d〈/mi〉〈mi〉μ〈/mi〉〈/math〉 (In certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When 〈em〉μ〈/em〉 is an ergodic hyperbolic measure, we prove that the exponential growth rate coicides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure 〈em〉ω〈/em〉, we also prove that there is a 〈em〉ω〈/em〉-full measured set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/math〉 such that for every 〈em〉f〈/em〉-invariant measure supported on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/math〉, the exponential growth rate equals to the free energy. And moreover, we prove that there is another 〈em〉ω〈/em〉-full measured set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/math〉 such that for every 〈em〉f〈/em〉-invariant measure supported on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/math〉, the exponential growth rate equals to the topological pressure.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Xinsheng Wang, Weisheng Wu, Yujun Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the 〈em〉u〈/em〉-equilibrium states are given.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Mohamed Sogoré, Chaker Jammazi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the problem of global finite-time stabilization of bilinear control systems by means of homogeneous feedback law is investigated. We prove under some reasonable assumptions on the operators 〈em〉A〈/em〉 and 〈em〉B〈/em〉 that continuous bounded and discontinuous unbounded feedbacks stabilize globally in finite-time the closed loop system. For illustrative, the examples of heat, Schrödinger and transport equations are considered where homogeneous stabilizing feedback is built for these systems.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Wen Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove that, in any infinite dimensional or asymptotic Teichmüller space, the angles between Teichmüller geodesic rays issuing from a common point, defined by using the law of cosines, do not always exist. As a consequence, any infinite dimensional or asymptotic Teichmüller space equipped with the Teichmüller metric is not a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mrow〉〈mi mathvariant="normal"〉CAT〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉κ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 space for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"〉〈mi〉κ〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉. We also establish a sufficient condition for the angles not to always exist in a Finsler manifold, and apply it to study the Hilbert metrics.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Tieshan He, Lang He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a new approach to studying a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave (i.e., (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/math〉)-sublinear) near zero and convex (i.e., (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/math〉)-superlinear) near ±∞. The reaction term is not assumed to be odd. We show that for all small values of the parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, the problem has infinitely many nodal solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Hong Chen, Yingjie Wang, Biqin Song, Han Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Magnitude-preserving ranking (MPRank) under Tikhonov regularization framework has shown competitive performance on information retrial besides theoretical advantages for computation feasibility and statistical guarantees. In this paper, we further characterize the learning rate and asymptotic bias of MPRank, and then propose a new debiased ranking algorithm. In terms of the operator representation and approximation techniques, we establish their convergence rates and bias characterizations. These theoretical results demonstrate that the new model has smaller asymptotic bias than MPRank, and can achieve the satisfactory convergence rate under appropriate conditions. In addition, some empirical examples are provided to verify the effectiveness of our debiased strategy.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Guoquan Qin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we establish the uniquely existence of the global mild solution to the nematic liquid crystal equations in Besov-Morrey spaces. Some self-similarity and large time behavior of the global mild solution are also investigated.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Philipp Holzinger, Ansgar Jüngel〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The large-time asymptotics of the density matrix solving a drift-diffusion-Poisson model for the spin-polarized electron transport in semiconductors is proved. The equations are analyzed in a bounded domain with initial and Dirichlet boundary conditions. If the relaxation time is sufficiently small and the boundary data is close to the equilibrium state, the density matrix converges exponentially fast to the spinless near-equilibrium steady state. The proof is based on a reformulation of the matrix-valued cross-diffusion equations using spin-up and spin-down densities as well as the perpendicular component of the spin-vector density, which removes the cross-diffusion terms. Key elements of the proof are time-uniform positive lower and upper bounds for the spin-up and spin-down densities, derived from the De Giorgi–Moser iteration method, and estimates of the relative free energy for the spin-up and spin-down densities.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Alexandre Kirilov, Wagner A.A. de Moraes〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator 〈em〉P〈/em〉 with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that 〈em〉P〈/em〉 is globally hypoelliptic. As an application, we obtain the characterization of global hypoellipticity on compact Lie groups and examples on the sphere and the torus. We also investigate relations between the global hypoellipticity of 〈em〉P〈/em〉 and global subelliptic estimates.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Vincenzo Ambrosio, Lorenzo Freddi, Roberta Musina〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we analyze the asymptotic behaviour of the Dirichlet fractional Laplacian 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/math〉, with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, on bounded domains in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/math〉 that become unbounded in the last 〈em〉k〈/em〉-directions. A dimension reduction phenomenon is observed and described via Γ-convergence.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Kengo Matsumoto〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a notion of 〈em〉λ〈/em〉-graph bisystem that consists of a pair 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of two labeled Bratteli diagrams 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈/math〉 satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of 〈em〉λ〈/em〉-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any 〈em〉λ〈/em〉-graph bisystem presents subshifts and conversely any subshift is presented by a 〈em〉λ〈/em〉-graph bisystem, called the canonical 〈em〉λ〈/em〉-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A 〈em〉λ〈/em〉-graph bisystem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 yields a pair of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebras written 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si145.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msubsup〉〈mo〉,〈/mo〉〈msubsup〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 that are first defined as the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebras of certain étale groupoids constructed from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. We study structure of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebras, and show that they are universal unital unique 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the 〈em〉λ〈/em〉-graph bisystem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. If a 〈em〉λ〈/em〉-graph bisystem comes from a 〈em〉λ〈/em〉-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 by its transition matrix 〈em〉A〈/em〉 of the graph, and the associated 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1118.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 is isomorphic to the Cuntz–Krieger algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈/math〉, whereas the other 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉L〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 is isomorphic to the crossed product 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈msub〉〈mrow〉〈mo〉⋊〈/mo〉〈/mrow〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈/mrow〉〈/msub〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉 of the commutative 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mi〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of continuous functions on the shift space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of the two-sided topological Markov shift by the automorphism 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si132.svg"〉〈msubsup〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 induced by the homeomorphism of the shift 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si133.svg"〉〈msub〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈/math〉. This phenomenon shows a duality between Cuntz–Krieger algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si128.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉O〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈/math〉 and the crossed product 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Λ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈msub〉〈mrow〉〈mo〉⋊〈/mo〉〈/mrow〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈/mrow〉〈/msub〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Xiao-song Liu, Tai-shun Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we first give the Bohr inequality of norm type for holomorphic mappings with lacunary series on the unit polydisk in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 under some restricted conditions. Next we also establish the Bohr inequality of norm type for holomorphic mappings with lacunary series on the unit ball of complex Banach spaces under some additional conditions, and the Bohr inequality of functional type for holomorphic mappings with lacunary series on the unit ball of complex Banach spaces. Our derived results reduce to the corresponding results in one complex variable.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Zhongrui Shi, Yu Wang, Qingying Bu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉A〈/em〉 be a norm bounded solid subset of a Banach lattice 〈em〉E〈/em〉 and 〈em〉n〈/em〉 be any positive integer. We prove that 〈em〉A〈/em〉 is an almost Dunford-Pettis set if and only if every positive weakly compact 〈em〉n〈/em〉-homogeneous polynomial from 〈em〉E〈/em〉 to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 maps 〈em〉A〈/em〉 to a relatively compact set in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Moreover, if 〈em〉E〈/em〉 is 〈em〉σ〈/em〉-Dedekind complete, we also prove that 〈em〉A〈/em〉 is an almost limited set if and only if every positive 〈em〉n〈/em〉-homogeneous polynomial from 〈em〉E〈/em〉 to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 maps 〈em〉A〈/em〉 to a relatively compact set in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Guiqiong Gong, Lan Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with the large-time behavior of the Cauchy problem to the 3d micropolar fluids in an infinite long flat nozzle domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉. In one dimensional case, this system tends time-asymptotically to the Navier–Stokes equations. That is to say, the basic wave patterns to the compressible micropolar fluids model are stable. Hence, in this paper we consider the nonlinear stability of planar rarefaction wave to the corresponding three dimensional model. Some cancellations on the flux terms and viscous terms are crucial. Moreover, a proper combining of damping term and rotation terms can provide an extra regularity of 〈em〉w〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Gerald Beer, M. Isabel Garrido〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real-valued Lipschitz function, the composition has this property. The key tools we use are the Efremovič lemma [21] and a theorem of Garrido and Jaramillo [22] that says that a function 〈em〉h〈/em〉 between metric spaces is Lipschitz if and only if whenever it is followed by a Lipschitz real-valued function in a composition, the composition is Lipschitz. We also present a streamlined proof of the Garrido-Jaramillo result itself, but one that still relies on their natural continuous linear operator from the Lipschitz space for the target space to the Lipschitz space for the domain. Separately, we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for Lipschitz-type functions as special cases.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Tingxi Hu, Lu Lu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the asymptotic properties of minimizers for a class of constraint minimization problems derived from the Maxwell-Schrödinger-Poisson system〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉⁎〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mspace width="0.25em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉〈/span〉 on the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-spheres 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉A〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="italic"〉λ〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉{〈/mo〉〈mi〉u〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈mo〉:〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉λ〈/mi〉〈mo stretchy="true" maxsize="2.4ex" minsize="2.4ex"〉}〈/mo〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msubsup〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msub〉〈/math〉 is the unique (up to translations) positive radial solution of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si61.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉. We prove that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msup〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉, then minimizers are relatively compact in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉A〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="italic"〉λ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉↗〈/mo〉〈mfrac〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉. On the contrary, if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈msup〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉, by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Mohammad Sadeghi, Nina Zorboska〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we prove that a Toeplitz operator with complex Borel measure symbol, whose total variation is Carleson, is weakly localized on the Bergman space. We define strongly localized and sufficiently localized operators on the Bergman space, and show that they are also weakly localized. Finally, we show that bounded Toeplitz operators with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉B〈/mi〉〈mi〉M〈/mi〉〈msup〉〈mrow〉〈mi〉O〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 symbols are strongly (and therefore also weakly) localized, and that the Bergman space Toeplitz algebra has yet another characterization as the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-algebra generated by this class of operators.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 486, Issue 1〈/p〉 〈p〉Author(s): Vladimir Kozlov, Jürgen Rossmann〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The paper deals with the Dirichlet problem for the nonstationary Stokes system in a cone. The authors obtain existence and uniqueness results for solutions in weighted Sobolev spaces and study the asymptotics of the solutions at infinity.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Anuj Abhishek〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉m〈/em〉 and 〈em〉n〈/em〉 be integers satisfying 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉m〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mi〉m〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉2〈/mn〉〈/math〉. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉M〈/mi〉〈mo〉,〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be a simple, real analytic, Riemannian manifold of dimension 〈em〉n〈/em〉 with boundary and 〈em〉f〈/em〉 be a rank 〈em〉m〈/em〉-tensor field defined over it. In this work, we prove a support theorem for the transverse ray transform of such tensor fields. More specifically, we prove that for a tensor field 〈em〉f〈/em〉 of rank 〈em〉m〈/em〉, if the transverse ray transform of 〈em〉f〈/em〉 vanishes over an appropriate open set of maximal geodesics of 〈em〉M〈/em〉, then the support of 〈em〉f〈/em〉 vanishes on the points of 〈em〉M〈/em〉 that lie on the union of the aforementioned open set of geodesics. We also show that if the tensor field is assumed to be symmetric, then one has a similar support theorem for the transverse ray transform of symmetric tensor fields of rank up to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Jean F. Barros, Eduardo S.G. Leandro〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We extend classical results on the localization of zeros of real univariate polynomials to the localization of zero sets of real multivariate polynomials 〈em〉P〈/em〉, more precisely, of real algebraic hypersurfaces (assuming 0 is a regular value). Through suitable changes of variables, we may verify whether such a hypersurface 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si158.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 intersects or not a given 〈em〉n〈/em〉-dimensional box 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉Π〈/mi〉〈/mrow〉〈mrow〉〈mi〉l〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉[〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉l〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉l〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉]〈/mo〉〈/math〉, and in the affirmative case, to locate with arbitrary precision the set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si148.svg"〉〈msup〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo〉∩〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="fraktur"〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉. Properties of the hypersurface such as being an analytic graph may also be deduced from our results, which include a non-differentiable, non-local version of the implicit function theorem for polynomials. Next, we apply the ideas of the first part to study the bifurcations of a one-parameter family of symmetric classes of relative equilibria of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉5〈/mn〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-body problem. The exact numbers of classes of relative equilibria are provided, and our technique allows for the localization of all relative equilibria.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 485, Issue 2〈/p〉 〈p〉Author(s): Dingyong Bai, Jianshe Yu, Meng Fan, Yun Kang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study complex dynamics of a non-autonomous predator-prey system with a Holling type II functional response and predator being generalist. We first studied basic dynamics such as boundedness, positive invariance, permanence, non-persistence and globally asymptotic stability. We also provide sufficient conditions for the existence, uniqueness and globally asymptotic stability of positive periodic solutions and boundary periodic solutions of the proposed model when parameters are periodic. We give the integral conditions to prove the extinction of prey and predator and globally asymptotic stability of boundary periodic solutions, and we show that these conditions have more reasonable biological interpretation than those expressed by supremum and infimum of parameters in some literature. We also perform numerical simulations to complement our analytical results and obtain more insights. Some of our numerical simulations indicate that periodic system may promote or suppress the permanence of its autonomous version with parameters being the averages of periodic parameters.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Huijie Qiao〈/p〉

Description: 〈p〉Publication date: 1 July 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications, Volume 487, Issue 1〈/p〉 〈p〉Author(s): Claudianor O. Alves, Romildo N. de Lima, Alânnio B. Nóbrega〈/p〉