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November  2007, 18(4): 613-626. doi: 10.3934/dcds.2007.18.613

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

Nils Ackermann 1, , Thomas Bartsch 2, and  Petr Kaplický 3,

1. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, Mexico
2. 

Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany
3. 

Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic

Received  November 2006 Revised  February 2007 Published  May 2007

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.
Keywords: Invariant Set, Singular Perturbation, Semilinear Parabolic Equation, Domain of Attraction., Connecting Orbit, Neumann Boundary Condition, Lusternik-Schnirelmann Category, Positive Equilibrium.
Mathematics Subject Classification: 35K55; 35K20, 35B25, 37B30, 55M3.
Citation: Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613
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