Superstable manifolds of semilinear parabolic problems.(English)Zbl 1129.35428

Summary: We investigate the dynamics of the semiflow $$\phi$$ induced on $$H_0^1(\Omega)$$ by the Cauchy problem of the semilinear parabolic equation $\partial_t u - \Delta u = f(x,u)$ on $$\Omega$$. 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 $\mathcal A := \left\{u \in H^1_0(\Omega) \mid \varphi^t(u) \rightarrow 0 \text{ as }t \rightarrow \infty\right\}$ 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.

MSC:

 35K20 Initial-boundary value problems for second-order parabolic equations 37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations 35K55 Nonlinear parabolic equations 37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems 47H20 Semigroups of nonlinear operators
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References:

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