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.