Opuscula Math. 41, no. 1 (2021), 25-53
https://doi.org/10.7494/OpMath.2021.41.1.25

 
Opuscula Mathematica

Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

Abderrahim Charkaoui
Houda Fahim
Nour Eddine Alaa

Abstract. We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

Keywords: variable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solution.

Mathematics Subject Classification: 35D30, 35K59, 35A01, 35K93, 35A16, 47H10.

Full text (pdf)

  1. B.E. Ainseba, B. Bendahmane, A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl. 9 (2008), 2086-2105.
  2. G. Akagi, K. Matsuura, Well-posedness and large-time behaviors of solutions for a~parabolic equation involving \(p(x)\)-Laplacian, The Eighth International Conference on Dynamical Systems and Differential Equations, a supplement volume of Discrete Contin. Dyn. Syst (2011), 22-31.
  3. N.E. Alaa, Solutions faibles d’équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise Pascal 3 (1996), no. 2, 1-15.
  4. N.E. Alaa, I. Mounir, Global existence for reaction-diffusion systems with mass control and critical growth with respect to the gradient, J. Math. Anal. Appl. 253 (2001), 532-557.
  5. N.E. Alaa, M. Pierre, Weak solutions for some quasi-linear elliptic equations with data measures, SIAM J. Math. Anal. 24 (1993), 23-35.
  6. T. Aliziane, M. Langlais, Degenerate diffusive SEIR model with logistic population control , Acta Math. Univ. Comenian., (N.S.) 75 (2006), no. 1, 185-198.
  7. M. Bendahmane, M. Saad, Mathematical analysis and pattern formation for a partial immune system modeling the spread of an epidemic disease, Acta Appl. Math. 115 (2011), 17-42.
  8. M. Bendahmane, P. Wittbold, A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\)-data, J. Differential Equations 249 (2010), no. 6, 1483-1515.
  9. L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), no. 6, 581-597.
  10. A. Charkaoui, N.E. Alaa, Weak periodic solution for semilinear parabolic problem with singular nonlinearities and \(L^1\) data, Mediterr. J. Math. 17 (2020), Article no. 108.
  11. A. Charkaoui, G. Kouadri, N.E. Alaa, Some results on the existence of weak periodic solutions for quasilinear parabolic systems with \(L^1\) data , Bol. Soc. Paran. Mat. (to appear).
  12. A. Charkaoui, G. Kouadri, O. Selt, N.E. Alaa, xistence results of weak periodic solution for some quasilinear parabolic problem with \(L^1\) data, An. Univ. Craiova Ser. Mat. Inform. 46 (2019), 66-77.
  13. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration , SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406.
  14. A. Elaassri, K. Lamrini Uahabi, A. Charkaoui, N.E. Alaa, S. Mesbahi, Existence of weak periodic solution for quasilinear parabolic problem with nonlinear boundary conditions, An. Univ. Craiova Ser. Mat. Inform. 46 (2019), 1-13.
  15. X. Fan, D. Zhao, On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m, p(x)}(\Omega)\), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446.
  16. H. Fahim, A. Charkaoui, N.E. Alaa, Weak solution for quasilinear parabolic systems with variable exponents and critical growth nonlinearities with respect to the gradient, J. Math. Inequal. (to appear).
  17. M. Fila, J. Lankeit, Lack of smoothing for bounded solutions of a semilinear parabolic equation, Adv. Nonlinear Anal. 9 (2020), no. 1, 1437-1452.
  18. Y. Fu, N. Pan, Existence of solutions for nonlinear parabolic problem with \(p(x)\)-growth, J. Math. Anal. Appl. 362 (2010), no. 2, 313-326.
  19. Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization, Commun. Contemp. Math. 20 (2018), no. 8, 1750065.
  20. O. Kovácik, J. Rákosník, On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\), Czechoslovak Math. J. 41 (1991), no. 4, 592-618.
  21. Z. Li, W. Gao, Existence of renormalized solutions to a nonlinear parabolic equation in ^1$ setting with nonstandard growth condition and gradient term , Math. Methods Appl. Sci. 38 (2015), 3043-3062.
  22. Z. Li, B. Yan, W. Gao, Existence of solutions to parabolic \(p(x)\)-Laplace equation with covection term via \(L^\infty\) estimates, Electron. J. Diff. Equ. 2015 (2015), no. 46, 1-21.
  23. M. Nakao, Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term, Opuscula Math. 39 (2019), no. 3, 395-414.
  24. S. Ouaro, A. Ouedraogo, Nonlinear parabolic problems with variable exponent and \(L^1\)-data, Electron. J. Differential Equations 32 (2017), 1-32.
  25. N. Papageorgiou, V. Radulescu, D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, 2019.
  26. A. Prignet, Existence and uniqueness of "entropy" solutions of parabolic problems with \(L^{1}\) data, Nonlinear Anal. 28 (1997), 1943-1954.
  27. V. Rădulescu Isotropic and anisotropic double-phase problems: old and new, Opuscula Math. 39 (2019), no. 2, 259-279.
  28. V. Rădulescu, D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Chapman and Hall/CRC, 2015.
  29. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science and Business Media, 2000.
  30. L. Shangerganesh, K. Balachandran, Solvability of reaction-diffusion model with variable exponents, Math. Methods Appl. Sci. 37 (2014), no. 10, 1436-1448.
  31. J. Simon, Compact sets in the space \(L^p(0,T;B)\), Ann. Mat. Pura Appl. 146 (1987), no. 4, 65-96.
  32. C. Zhang, S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and \(L^1\) data, J. Differential Equations 248 (2010), no. 6, 1376-1400.
  • Abderrahim Charkaoui (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-1425-7248
  • Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
  • Houda Fahim
  • Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
  • Nour Eddine Alaa
  • ORCID iD https://orcid.org/0000-0001-8169-8663
  • Laboratory LAMAI, Faculty of Science and Technology of Marrakech, B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2020-09-29.
  • Revised: 2020-12-09.
  • Accepted: 2020-12-09.
  • Published online: 2021-02-08.
Opuscula Mathematica - cover

Cite this article as:
Abderrahim Charkaoui, Houda Fahim, Nour Eddine Alaa, Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent, Opuscula Math. 41, no. 1 (2021), 25-53, https://doi.org/10.7494/OpMath.2021.41.1.25

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.