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

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.

### 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)
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