Large time behaviour for the heat equation on $\mathbb{Z},$ moments and decay rates

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 $\mathbb{Z}$ on ${\ell }^p$ 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 ${\ell }^p$ decay results for the fundamental solution. We use that estimates to get rates on the ${\ell }^p$ decay and large time behaviour of solutions. For the ${\ell }^2$ case, we get optimal decay by use of Fourier techniques.

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