Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 613 - 626, Volume 18, Issue 4, August 2007      doi:10.3934/dcds.2007.18.613

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Nils Ackermann - Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, Mexico (email)
Thomas Bartsch - Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany (email)
Petr Kaplický - Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic (email)

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.

Keywords:  Semilinear Parabolic Equation, Singular Perturbation, Neumann Boundary Condition, Lusternik-Schnirelmann Category, Invariant Set, Connecting Orbit, Positive Equilibrium, Domain of Attraction.
Mathematics Subject Classification:  35K55; 35K20, 35B25, 37B30, 55M30.

Received: November 2006;      Revised: February 2007;      Available Online: May 2007.