Journal of Dynamics and Differential Equations

, Volume 17, Issue 1, pp 115–173 | Cite as

Superstable Manifolds of Semilinear Parabolic Problems

Article

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 Wk’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.

Keywords

Invariant manifolds connecting orbits nodal properties 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematisches InstitutJustus-Liebig-UniversitätGießenGermany

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