Large time behaviour for the heat equation on , moments and decay rates
arxiv(2024)
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 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.
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