Gradient-like parabolic semiflows on $BUC({\mathbb{R}}^N)$

Abstract

We prove that a class of weighted semilinear reaction diffusion equations on ${\mathbb{R}}^N$ generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on ${\mathbb{R}}^N$. If $N=1$ we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.