An invariant set generated by the domain topology for parabolic semiflows with small diffusion

Nils Ackermann; Thomas Bartsch; Petr Kaplický

doi:10.3934/dcds.2007.18.613

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2007, Volume 18, Issue 4: 613-626. Doi: 10.3934/dcds.2007.18.613

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An invariant set generated by the domain topology for parabolic semiflows with small diffusion

1.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510
2.
Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen
3.
Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8

Received: November 2006

Revised: February 2007

Published: October 2007

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Abstract

We consider the singularly perturbed semilinear parabolic problem $u_t-d^2\Delta u+u=f(u)$ with homogeneous Neumann boundary conditions on a smoothly bounded domain $\Omega\subseteq \mathbb{R}^N$. Here $f$ is superlinear at $0$, and $\pm\infty$ and has subcritical growth. For small $d>0$ we construct a compact connected invariant set $X_d$ in the boundary of the domain of attraction of the asymptotically stable equilibrium $0$. The main features of $X_d$ are that it consists of positive functions that are pairwise non-comparable, and its topology is at least as rich as the topology of $\partial\Omega$ in a certain sense. If the number of equilibria in $X_d$ is finite, then this implies the existence of connecting orbits within $X_d$ that are not a consequence of a well known result by Matano.

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Mathematics Subject Classification: 35K55; 35K20, 35B25, 37B30, 55M30.

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