Abstract
ABSTRACT We consider the semilinear parabolic equation on , assuming that f is an arbitrary C 1 function satisfying and . We prove that any bounded positive solution that decays to zero at spatial infinity, uniformly with respect to t, converges to a (single) stationary solution as . Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as , approaches a set of steady states that are radially symmetric about a common origin in . To this aim we introduce a new tool that we call first moments of energy. Having established the symmetrization, we apply a general convergence result for gradient-like dynamical systems. This amounts to showing that the dimension of the kernel of the linearized operator around an equilibrium w matches the dimension of a manifold of equilibria passing through w.
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