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

Existence of stable and unstable periodic solutions for semilinear parabolic problems

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

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.

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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