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Superstable Manifolds of Semilinear Parabolic Problems

Authors:

Email author" target="_blank">Nils?Ackermann Email author Thomas?Bartsch

Institution:

(1) Mathematisches Institut, Justus-Liebig-Universität, Arndtstr. 2, D-35392 Gießen, Germany

Abstract:

We investigate the dynamics of the semiflow φ induced on H₀¹(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.

Keywords:

Invariant manifolds connecting orbits nodal properties

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