##
**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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\textit{N. Ackermann} and \textit{T. Bartsch}, J. Dyn. Differ. Equations 17, No. 1, 115--173 (2005; Zbl 1129.35428)

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### References:

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