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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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