Self-similar behaviour of a non-local diffusion equation with time delay
Dynamical Systems
2019-01-01 v1 Analysis of PDEs
Functional Analysis
Probability
Abstract
We study the asymptotic behaviour of solutions of a class of linear non-local measure-valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic like behaviour in the large times, that is precisely expressed in term of heat kernel. Our proof relies on the study of a-self-similar-rescaled family of solutions. We first identify the asymptotic behaviour of the solutions by deriving a convergence result in the sense of the Young measures. Then we strengthen this convergence by deriving suitable fractional Sobolev compactness estimates. As a by-product, our main result allows to obtain asymptotic results for a class of piecewise constant stochastic processes with memory.
Cite
@article{arxiv.1812.11342,
title = {Self-similar behaviour of a non-local diffusion equation with time delay},
author = {Arnaud Ducrot and Alexandre Genadot},
journal= {arXiv preprint arXiv:1812.11342},
year = {2019}
}