Large time behaviour for the heat equation on Z, moments and decay rates

Abstract

Abstract The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on l p spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and l p decay results for the fundamental solution. We use that estimates to get rates on the l p decay and large time behaviour of solutions. For the l 2 case, we get optimal decay by use of Fourier techniques.