, Volume 17, Issue 1, pp 115-173

Superstable Manifolds of Semilinear Parabolic Problems

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Abstract

We investigate the dynamics of the semiflow φ induced on H 0 1(Ω) by the Cauchy problem of the semilinear parabolic equation $$\partial_{t}u - \Delta u = f(x, u)$$ on Ω. Here $\Omega \subseteq \mathbb{R}^{N}$ is a bounded smooth domain, and $f: \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ has subcritical growth in u and satisfies $f (x, 0) \equiv 0$ . In particular we are interested in the case when f is definite superlinear in u. The set $$ {\cal A}: = \{u \in H^{1}_{0} (\Omega ) | \varphi^{t} (u) \rightarrow 0 \hbox{as} t \rightarrow \infty\} $$ of attraction of 0 contains a decreasing family of invariant sets $$ W_{1} \supseteq W_{2} \supseteq W_{3} \supseteq \ldots $$ distinguished by the rate of convergence $\varphi^{t} (u) \rightarrow 0$ . We prove that the W k ’s are global submanifolds of $H^{1}_{0} (\Omega)$ , and we find equilibria in the boundaries $\overline{W}_{k} \backslash W_{k}$ . We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.

Supported by DFG Grant BA 1009/15-1.