A shock layer arising as the source term collapses in the p(x)-Laplacian equation

Standard

A shock layer arising as the source term collapses in the p(x)-Laplacian equation. / Antontsev, S. N.; Kuznetsov, I.; Sazhenkov, S. A.

In: Problemy Analiza, Vol. 9(27), No. 3, 2020, p. 31-53.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{7c7021834db7412a8a7797fddcfe8cb5,

title = "A shock layer arising as the source term collapses in the p(x)-Laplacian equation",

abstract = "We study the Cauchy–Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term. 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 {\textquoteleft}outer{\textquoteright} 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.",

keywords = "parabolic equation, nonstandard growth, variable non-linearity, non-instantaneous impulse, energy solution, shock layer",

author = "Antontsev, {S. N.} and I. Kuznetsov and Sazhenkov, {S. A.}",

year = "2020",

doi = "10.15393/j3.art.2020.8990",

language = "English",

volume = "9(27)",

pages = "31--53",

journal = "Problemy Analiza",

issn = "2306-3424",

publisher = "Petrozavodsk State University",

number = "3",

}

RIS

TY - JOUR

T1 - A shock layer arising as the source term collapses in the p(x)-Laplacian equation

AU - Antontsev, S. N.

AU - Kuznetsov, I.

AU - Sazhenkov, S. A.

PY - 2020

Y1 - 2020

N2 - We study the Cauchy–Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term. 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.

AB - We study the Cauchy–Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term. 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.

KW - parabolic equation

KW - nonstandard growth

KW - variable non-linearity

KW - non-instantaneous impulse

KW - energy solution

KW - shock layer

UR - http://www.scopus.com/inward/record.url?scp=85101496368&partnerID=8YFLogxK

U2 - 10.15393/j3.art.2020.8990

DO - 10.15393/j3.art.2020.8990

M3 - Article

VL - 9(27)

SP - 31

EP - 53

JO - Problemy Analiza

JF - Problemy Analiza

SN - 2306-3424

IS - 3

ER -

ID: 27361442