Abstract: The Cauchy--Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term is studied. The minor term depends on a small parameter ε>0 and, as ε→0, converges weakly* to the expression incorporating the Dirac delta function, which models a shock (impulsive) loading. We establish that the shock layer, associated with the Dirac delta function, is formed as ε→0, and that the family of weak solutions of the original problem converges to a solution of a two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial and boundary conditions, so that the `outer' macroscopic solution beyond the shock layer is governed by the usual homogeneous p(x)-Laplacian equation, while the shock layer solution is defined on the microscopic level and obeys the ordinary differential equation derived from the microstructure of the shock layer profile.

Cite: Antontsev S.N. , Kuznetsov I.V. , Sazhenkov S.A.
A shock layer arising as the source term collapses in the p(x)-Laplacian equation
Problemy Analiza. 2020. V.27. N3. P.31-53. DOI: 10.15393/j3.art.2020.8990 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Aug 25, 2020
Accepted:	Oct 28, 2020
Published online:	Nov 3, 2022

Identifiers:

Web of science:	WOS:000590954400003
Scopus:	2-s2.0-85101496368
Elibrary:	44283108
OpenAlex:	W3109740896

Citing:

DB	Citing
Scopus	6
OpenAlex	6
Elibrary	8
Web of science	6

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