Gradient-like parabolic semiflows on BUC(R-N)

We prove that a class of weighted semilinear reaction diffusion equations o n R-N generates gradient-like semiflows on the Banach space of bounded unif ormly continuous functions on R-N. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential dec ay of the stationary solutions, which is obtained by means of a decay estim ate of the kernel of the underlying semigroup.