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

Proceedings of the Royal Society of Edinburgh 128A(1998), 1281–1291
Original available at doi:S0308210500027323
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.