ISSN:
1572-9222
Keywords:
Nonlinear elliptic equations
;
cylindrical domains
;
infinitedimensional dynamical systems
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x 0,x 1,⋯,x n )∃(s 0, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} \\ {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } \\ {|u(x)|globallybounded} \\ \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial valueϕ, in the spaceC 0 0 $$(\overline {\Omega '} )$$ and satisfying the strong order-preserving property. In the case thata ij andf do not depend onx 0 or are periodic inx 0, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx 0 tends to infinity. Proofs are strongly based on maximum and comparison techniques.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02227486
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