Geodesic
A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$-Laplacian equation
Problemy analiza, Tome 9 (2020) no. 3, pp. 31-53.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the Cauchy–Dirichlet problem for the $p(\boldsymbol{x})$-Laplacian equation with a regular finite nonlinear minor term. The minor term depends on a small parameter $\varepsilon>0$ and, as $\varepsilon\to 0$, converges weakly$^\star$ 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 $\varepsilon\to 0$, 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 $p(\boldsymbol{x})$-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: nonstandard growth, variable nonlinearity, non-instantaneous impulse, energy solution, shock layer.
Mots-clés : parabolic equation 
@article{PA_2020_9_3_a2,
     author = {S. N. Antontsev and I. V. Kuznetsov and S. A. Sazhenkov},
     title = {A shock layer arising as the source term collapses in the $p(\boldsymbol{x})${-Laplacian} equation},
     journal = {Problemy analiza},
     pages = {31--53},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2020_9_3_a2/}
}
TY  - JOUR
AU  - S. N. Antontsev
AU  - I. V. Kuznetsov
AU  - S. A. Sazhenkov
TI  - A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$-Laplacian equation
JO  - Problemy analiza
PY  - 2020
SP  - 31
EP  - 53
VL  - 9
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2020_9_3_a2/
LA  - en
ID  - PA_2020_9_3_a2
ER  - 
%0 Journal Article
%A S. N. Antontsev
%A I. V. Kuznetsov
%A S. A. Sazhenkov
%T A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$-Laplacian equation
%J Problemy analiza
%D 2020
%P 31-53
%V 9
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2020_9_3_a2/
%G en
%F PA_2020_9_3_a2
S. N. Antontsev; I. V. Kuznetsov; S. A. Sazhenkov. A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$-Laplacian equation. Problemy analiza, Tome 9 (2020) no. 3, pp. 31-53. http://geodesic.mathdoc.fr/item/PA_2020_9_3_a2/

[1] Agarwal R., Hristova S., O'Regan D., Non-Instantaneous Impulses in Differential Equations, Springer, 2017 | DOI | MR | Zbl

[2] Alkhutov Y. A., Zhikov V. V., “Existence theorems for solutions of parabolic equations with a variable order of nonlinearity”, Tr. Mat. Inst. Steklova, 270, 2010, 21–32 | DOI | MR | Zbl

[3] Alkhutov Y. A., Zhikov V. V., “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent”, Sbornik: Mathematics, 205:3 (2014), 307–318 | DOI | MR | Zbl

[4] Antontsev S., Kuznetsov I., Shmarev S., “Global higher regularity of solutions to singular $p(x, t)$-parabolic equations”, J. Math. Anal. Appl., 466:1 (2018), 238–263 | DOI | MR | Zbl

[5] Antontsev S., Shmarev S., Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up, Atlantis Press, 2015 | DOI | MR | Zbl

[6] Coutinho F. A. B., Nogami Y., Toyama F. M., “Unusual situations that arise with the Dirac delta function and its derivative”, Revista Brasileira de Ensino de Física, 31:4 (4302) (2009), 1–7 | DOI | MR

[7] Diening L., Nägele P., Ružička M., “Monotone operator theory for unsteady problems in variable exponent spaces”, Complex Var. Elliptic Equ., 57:11 (2012), 1209–1231 | DOI | MR | Zbl

[8] Griffiths D., Walborn S., “Dirac deltas and discontinuous functions”, Amer. J. Phys., 67 (1999), 446 | DOI | MR | Zbl

[9] Hanche-Olsen H., Holden H., “The Kolmogorov-Riesz compactness theorem”, Expositiones Math., 28 (2010), 385–394 | DOI | MR | Zbl

[10] Kreyszig E., Advanced Engineering Mathematics, 10th edition, John Wiley Sons Inc, 2011 | DOI | MR | Zbl

[11] Miller B. M., Rubinovich E. Ya., Impulsive Control in Continuous and Discrete-Continuous Systems, Springer, 2003 | DOI | MR

[12] Perthame B., Kinetic Formulations of Conservation Laws, Oxford Univ. Press, Oxford, 2002 | MR

[13] Plotnikov P. I., Sazhenkov S. A., “Kinetic formulation for the Graetz-Nusselt ultra-parabolic equation”, J. Math. Anal. Appl., 304 (2005), 703–724 | DOI | MR | Zbl

[14] Sazhenkov S. A., “Noisiness estimate for kinetic solutions of scalar conservation laws”, Collection of scientific works (electronic resource), International Conference ‘Lomonosov’s Readings on Altai: Fundamental Problems of Science and Technology – 2018', ed. Rodionov E. D., Altai State University Press, Barnaul, 2018, 249–257

[15] Serrin J., “Mathematical Principles of Classical Fluid Mechanics”, Fluid Dynamics I, Encyclopedia of Physics, 3/8/1, ed. Truesdell C., Springer, Berlin–Heidelberg, 1959, 125–263 | DOI | MR

[16] Temam R., Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, 2000 | DOI | MR

[17] Vasseur A., “Well-posedness of scalar conservation laws with singular sources”, Methods Appl. Anal., 9:2 (2002), 291–312 | DOI | MR | Zbl

[18] Wang J., Fečkan M., Non-Instantaneous Impulsive Differential Equations. Basic Theory and Computation, IOP Publishing, 2018 | DOI