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Self-similar behaviour of a non-local diffusion equation with time delay

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.
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Submitted on : Friday, December 21, 2018 - 10:23:02 PM
Last modification on : Wednesday, February 2, 2022 - 3:54:38 PM
Long-term archiving on: : Friday, March 22, 2019 - 4:51:09 PM


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  • HAL Id : hal-01964356, version 1
  • ARXIV : 1812.11342


Arnaud Ducrot, Alexandre Genadot. Self-similar behaviour of a non-local diffusion equation with time delay. 2018. ⟨hal-01964356⟩



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