Convergence to equilibrium for semilinear parabolic problems in ℝⁿ

We consider the semilinear parabolic equation u_t = Δu +f(u) on ℝ^N, assuming that f is an arbitrary C¹ function satisfying f(0) = 0 and f′(0) < 0. 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 t → ∞. Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as t → ∞, u(x, t) approaches a set of steady states that are radially symmetric about a common origin in ℝ^N. To this aim we introduce a new toot 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.