##
**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:

[1] | Ackermann N., Bartsch T., Kaplicky P., Quittner P. A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems. Submitted. · Zbl 1143.37049 |

[2] | Amann H. (1993). Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), Vol. 133 of Teubner-Texte Math., Teubner, Stuttgart. pp 9–126. |

[41] | Rabinowitz, P.H. (1986). Minimax methods in critical point theory with applications to differential equations, Vol. 65 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC. · Zbl 0609.58002 |

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