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Time periodic solutions of a class of degenerate parabolic equations

Time periodic solutions of a class of degenerate parabolic equations This paper is devoted to the time periodic solutions to the degenerate parabolic equations of the form $$\frac{{\partial u}}{{\partial t}} = \vartriangle u^m + u^p (a(x,t) - b(x,t)u) in \Omega \times R$$ under the Dirichlet boundary value condition, wherem>1,p≥0, Ω∈R N is a bounded domain with smooth boundary σΩ anda,b are positive, smooth functions which are periodic int with period ω>0. The existence of nontrivial nonnegative solutions is established provided that 0≤p<m. The existence is also proved in the casep=m but with an additional assumption $$\begin{array}{*{20}c} {\min } \\ {\bar Q} \\ \end{array} a(x,t) > \lambda _1 $$ , where $$\lambda _1 $$ is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Time periodic solutions of a class of degenerate parabolic equations

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References (21)

Publisher
Springer Journals
Copyright
Copyright © 2000 by Science Press
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02677678
Publisher site
See Article on Publisher Site

Abstract

This paper is devoted to the time periodic solutions to the degenerate parabolic equations of the form $$\frac{{\partial u}}{{\partial t}} = \vartriangle u^m + u^p (a(x,t) - b(x,t)u) in \Omega \times R$$ under the Dirichlet boundary value condition, wherem>1,p≥0, Ω∈R N is a bounded domain with smooth boundary σΩ anda,b are positive, smooth functions which are periodic int with period ω>0. The existence of nontrivial nonnegative solutions is established provided that 0≤p<m. The existence is also proved in the casep=m but with an additional assumption $$\begin{array}{*{20}c} {\min } \\ {\bar Q} \\ \end{array} a(x,t) > \lambda _1 $$ , where $$\lambda _1 $$ is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 6, 2007

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