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

Treść / Zawartość
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Strony
25--53
Opis fizyczny
Bibliogr. 32 poz.
Twórcy
  • Laboratory LAMAI Faculty of Science and Technology of Marrakech B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
autor
  • Laboratory LAMAI Faculty of Science and Technology of Marrakech B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
  • Laboratory LAMAI Faculty of Science and Technology of Marrakech B.P. 549, Av. Abdelkarim Elkhattabi, 40 000, Marrakech, Morocco
Bibliografia
  • [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, Wel l-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’equations paraboliques quasi-lineaires avec donnees 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 el liptic 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 L1 -data, J. Differential Equations 249 (2010), no. 6, 1483-1515.
  • [9] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to el liptic 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 L1 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 L1 data, Bol. Soc. Paran. Mat. (to appear).
  • [12] A. Charkaoui, G. Kouadri, O. Selt, N.E. Alaa, Existence results of weak periodic solution for some quasilinear parabolic problem with L1 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 Lp(x)(L) and Wmp(x)(fi), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446.
  • [16] H. Fahim, A. Charkaoui, N.E. Alaa, M. Guedda, 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] J. Giacomoni, V. Radulescu, G. Warnault, Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization, Commun. Contemp. Math. 20 (2018), no. 8, 1750065.
  • [20] O. Kovacik, J. Rakosmk, On spaces Lp(x) and Wk,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 L1 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 LTO 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 quasi-linear 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 L1 -data, Electron. J. Differential Equations 32 (2017), 1-32.
  • [25] N. Papageorgiou, V. Radulescu, D. Repovs, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, 2019.
  • [26] A. Prignet, Existence and uniqueness of “entropy” solutions of parabolic problems with L1 data, Nonlinear Anal. 28 (1997), 1943-1954.
  • [27] V. Radulescu, Isotropic and anisotropic double-phase problems: old and new, Opuscula Math. 39 (2019), no. 2, 259-279.
  • [28] V. Radulescu, D. Repovs, 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 & 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 Lp (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 L1 data, J. Differential Equations 248 (2010), no. 6, 1376-1400.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-d606e99b-fc01-45f8-b307-b3c94988a213
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