Numerical analysis of nonlinear parabolic problems with variable exponent and \(L^1\) data

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DOI:

https://doi.org/10.56754/0719-0646.2402.0187

Abstract

In this paper, we make the numerical analysis of the mild solution which is also an entropy solution of parabolic problem involving the \(p(x)-\)Laplacian operator with \(L^1-\) data.

Keywords

elliptic-parabolic , numerical iterative method , variable exponent , mild solution , renormalized solution
  • Stanislas Ouaro Laboratoire de Mathématiques et Informatique (LAMI), Unité de Formation et de Recherche en Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP. 7021 Ouagadougou 03, Burkina Faso.
  • Noufou Rabo Laboratoire de Mathématiques et Informatique (LAMI), Unité de Formation et de Recherche en Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP. 7021 Ouagadougou 03, Burkina Faso.
  • Urbain Traoré Laboratoire de Mathématiques et Informatique (LAMI), Unité de Formation et de Recherche en Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP. 7021 Ouagadougou 03, Burkina Faso.
  • Pages: 187–209
  • Date Published: 2022-08-22
  • Vol. 24 No. 2 (2022)

S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions”, Nonlinear Anal., vol. 60, no. 3, pp. 515–545, 2005. DOI: https://doi.org/10.1016/s0362-546x(04)00393-1

S. N. Antontsev and V. Zhikov, “Higher integrability for parabolic equations of p(x, t)- Laplacian type”, Adv. Differential Equations, vol. 10, no. 9, pp. 1053–1080, 2005.

M. Bendahmane, K. H. Karlsen and M. Saad, “Nonlinear anisotropic elliptic and parabolic equations with variable exponents and L1 data”, Commun. Pure Appl. Anal., vol. 12, no. 3, pp. 1201–1220, 2013. DOI: https://doi.org/10.3934/cpaa.2013.12.1201

M. Bendahmane and P. Wittbold and A. Zimmermann, “Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1−data”, J. Differential Equations, vol. 249, no. 6, pp. 1483–1515, 2010. DOI: https://doi.org/10.1016/j.jde.2010.05.011

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, “An L1- theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 22, no. 2, pp. 241–273, 1995.

Ph. Bénilan and M. G. Crandall and A. Pazy, Evolution equations governed by accretive operators, unpublished book.

A. E. Berger, H. Brézis and J. C. W. Rogers, “A numerical method for solving the problem ut − ∆f (u) = 0, RAIRO Anal. Numér., vol. 13, no. 4, pp. 297–312, 1979. DOI: https://doi.org/10.1051/m2an/1979130402971

L. C. Berselli, D. Breit and L. Diening, “Convergence analysis for a finite element approximation of a steady model for electrorheological fluids”, Numer. Math., vol. 132, no. 4, pp. 657–689, 2016.

D. Blanchard and F. Murat, “Renormalised solutions of nonlinear parabolic problems with L1−data: existence and uniqueness”, Proc. Roy. Soc. Edinburgh Sect. A, vol. 127, no. 6, pp. 1137–1152, 1997.

D. Breit and L. Diening and S. Schwarzacher, “Finite element approximation of the p(·)- Laplacian”, SIAM J. Numer. Anal., vol. 53, no. 1, pp. 551–572, 2015.

D. Breit and P. R. Mensah, “Space-time approximation of parabolic systems with variable growth”, IMA J. Numer. Anal., vol. 40, no. 4, pp. 2505–2552, 2020.

M. Caliari and S. Zuccher, “The inverse power method for the p(x)-Laplacian problem”, J. Sci. Comput., vol. 65, no. 2, pp. 698–714, 2015.

M. Caliari and S. Zuccher, “Quasi-Newton minimization for the p(x)-Laplacian problem”, J. Comput. Appl. Math., vol. 309, pp. 122–131, 2017.

Y. Chen, S. Levine and M. Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math., vol. 66, no. 4, pp. 1383–1406, 2006.

L. Diening, P. Harjulehto, P. Hästö and M. RůžiÄka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Heidelberg: Springer, 2011.

L. Diening, P. Nägele and M. RůžiÄka, “Monotone operator theory for unsteady problems in variable exponent spaces”, Complex Var. Elliptic Equ., vol. 57, no. 11, pp. 1209–1231, 2012.

Z. Dou, K. Gao, B. Zhang, X. Yu, L. Han and Z. Zhu, “Realistic image rendition using a variable exponent functional model for retinex”, Sensors, vol. 16, no. 6, 16 pages, 2016.

W. Jäger and J. KaÄur, “Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes”, RAIRO Modél. Math. Anal. Numér., vol. 29, no. 5, pp. 605–627, 1995.

F. Karami, K. Sadik and L. Ziad, “A variable exponent nonlocal p(x)-Laplacian equation for image restoration”, Comput. Math. Appl., vol. 75, no. 2, pp. 534–546, 2018.

J. KaÄur, “Solution of some free boundary problems by relaxation schemes”, SIAM J. Numer. Anal., vol. 36, no. 1, pp. 290–316, 1999.

O. KováÄik and J. Rákosník, “On spaces Lp(x) and Wk, p(x)”, Czechoslovak Math. J., vol. 41, no. 4, pp. 592–618, 1991.

E. Magenes, R. H. Nochetto and C. Verdi, “Energy error estimates for a linear scheme to approximate nonlinear parabolic problems”, RAIRO Modél. Math. Anal. Numér., vol. 21, no. 4, pp. 655–678, 1987.

E. Maitre, “Numerical analysis of nonlinear elliptic-parabolic equations”, M2AN Math. Model. Numer. Anal., vol. 36, no. 1, pp. 143–153, 2002.

S. Ouaro and A. Ouédraogo, “Nonlinear parabolic problems with variable exponent and L1−data”, Electron. J. Differential Equations, Paper No. 32, 32 pages, 2017.

S. Ouaro and S. Traoré, “Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth”, Int. J. Evol. Equ., vol. 4, no. 4, pp. 451–471, 2010.

L. M. Del Pezzo, A. L. Lombardi and S. Martínez, “Interior penalty discontinuous Galerkin FEM for the p(x)-Laplacian”, SIAM J. Numer. Anal., vol. 50, no. 5, pp. 2497–2521, 2012.

M. RůžiÄka, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Berlin: Springer-Verlag, 2000.

V. D. Rădulescu and D. D. Repovš, Partial differential equations with variable exponents, Monographs and Research Notes in Mathematics, Boca Raton: CRC Press, 2015.

C. Zhang and S. Zhou, “Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data”, J. Differential Equations, vol. 248, no. 6, pp. 1376–1400, 2010.

V. V. Zhikov, “On the density of smooth functions in Sobolev-Orlicz spaces”, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), vol. 310, pp. 67–81, 2004.

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Published

2022-08-22

How to Cite

[1]
S. Ouaro, N. Rabo, and U. Traoré, “Numerical analysis of nonlinear parabolic problems with variable exponent and \(L^1\) data”, CUBO, vol. 24, no. 2, pp. 187–209, Aug. 2022.

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