An invariant set generated by the domain topology for parabolic semiflows with small diffusion. (English) Zbl 1139.35053

Ackermann, Nils; Bartsch, Thomas; Kaplický, Petr

Discrete Contin. Dyn. Syst. 18, No. 4, 613-626 (2007).

The authors deal with a 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 diffusion, i.e., \(d\ll 1\), a compact connected invariant set \(X_d\) in the boundary of the domain of attraction of the asymptotically stable equilibrium \(0\) is constructed. The main features of \(X_d\) are that it consists of positive functions that are pairwise non-comparable, and in a certain sense its topology is at least as rich as the topology on the boundary \(\partial\Omega\). If \(X_d\) contains finitely many equilibria, then this implies the existence of connecting orbits within \(X_d\), which are not a consequence of Matano’s well-known result.
The proofs include topological arguments based on tools like Lusternik-Schnirelmann category and Alexander-Spanier (or Čech) cohomology.

Reviewer: Christian Pötzsche (München)

MSC:

35K55	Nonlinear parabolic equations
35K20	Initial-boundary value problems for second-order parabolic equations
35B25	Singular perturbations in context of PDEs
37B30	Index theory for dynamical systems, Morse-Conley indices
55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)

Keywords:

Neumann boundary condition; Lusternik-Schnirelmann category; invariant set; connecting orbit; positive equilibrium; domain of attraction