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Existence of stable and unstable periodic solutions for semilinear parabolic problems

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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.

    Mathematics Subject Classification: Primary: 35K20, 35B10.


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