Abstract: We study the Cauchy problem with periodic initial data for the heat equation, which includes a nonlocal in time integral term. This term models a fading memory effect and has the form of convolution of a nonlinear function depending on solution with a smooth relaxation kernel. The kernel contains a small parameter 𝜀 > 0 and, as 𝜀 → 0 , collapses to the Dirac delta function supported at some given moment of time t = τ. The Dirac delta function, in turn, models a shock (impulsive) loading at the moment t = τ . We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as 𝜀 → 0 , and the family of weak solutions of the considered problem converges to a solution of the two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the classical homogeneous heat equation, while the transition layer solution is defined on the microscopic level and obeys the Volterra integro-differential equation derived from the microstructure of the relaxation profile.

Cite: Kuznetsov I. , Sazhenkov S.
The impulsive heat equation with the Volterra transition layer
Journal of Elliptic and Parabolic Equations. 2022. V.8. N2. P.959-993. DOI: 10.1007/s41808-022-00182-9 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Sep 13, 2021
Accepted:	Aug 14, 2022
Published online:	Sep 10, 2022

Identifiers:

Web of science:	WOS:000852367200002
Scopus:	2-s2.0-85137786156
Elibrary:	51636155
OpenAlex:	W4296270210

Citing:

DB	Citing
Scopus	2
OpenAlex	2
Elibrary	3
Web of science	2

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