Existence of stable and unstable periodic solutions for semilinear parabolic problems

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April 1997, 3(2): 207-216. doi: 10.3934/dcds.1997.3.207

Existence of stable and unstable periodic solutions for semilinear parabolic problems

E. N. Dancer ^1, and Norimichi Hirano ^2,

1.	School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
2.	Department of Mathematics, Yokohama National University, 156 Tokiwadai Hodogaya-ku - Yokohama

Received October 1996 Published January 1997

Abstract
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In this paper, we show the existence of stable and unstable $T-$periodic solutions for a semilinear parabolic equation

$\frac{\partial u}{\partial t} - \Delta u = g(x,u) + h( t, x ),\quad \text{in} \quad (0,T) \times \Omega$

$u=0 ,\quad \text{on}\quad (0,T) \times \partial \Omega$

$u(0) = u(T),\quad \text{in} \quad \overline \Omega$

where $\Omega \subset R^N$ is a bounded domain with a smooth boundary, $g:\overline{\Omega} \times R \rightarrow R$ is a continuous function such that $g(x,\cdot )$ has a superlinear growth for each $x \in \overline{\Omega} $ and $h:(0,T) \times \Omega \to R$ is a continuous function.

Keywords: Stable and unstable solution, semilinear parabolic problems..

Mathematics Subject Classification: Primary: 35K20, 35B1.

Citation: E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207

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