Superstable Manifolds of Semilinear Parabolic Problems

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.