Received October 10, 2016. First published October 17, 2017.
Abstract: We prove the existence of solutions to nonlinear parabolic problems of the following type: $ \dfrac{\partial b(u)}{\partial t}+ A(u) = f + {\rm div}(\Theta(x; t; u))& \text{in} Q, u(x; t) = 0 & \text{on} \partial\Omega\times[0; T], b(u)(t = 0) = b(u_0) & \text{on} \Omega,$ where $b \Bbb{R}\to\Bbb{R}$ is a strictly increasing function of class ${\mathcal C}^1$, the term $ A(u) = -{\rm div} (a(x, t, u,\nabla u)) $ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta\colon\Omega\times[0; T]\times\Bbb{R}\to\Bbb{R}$ is a Carathéodory, noncoercive function which satisfies the following condition: $\sup_{|s|\le k} |\Theta({\cdot},{\cdot},s)| \in E_{\psi}(Q)$ for all $k > 0$, where $\psi$ is the Musielak complementary function of $\Theta$, and the second term $f$ belongs to $L^1(Q)$.
References: [1] A. Aberqi, J. Bennouna, M. Mekkour, H. Redwane: Existence results for a nonlinear parabolic problems with lower order terms. Int. J. Math. Anal., Ruse 7 (2013), 1323-1340. DOI 10.12988/ijma.2013.13130 | MR 3053336 | Zbl 1284.35216 [2] M. L. Ahmed Oubeid, A. Benkirane, M. Sidi El Vally: Nonlinear elliptic equations involving measure data in Musielak-Orlicz-Sobolev spaces. J. Abstr. Differ. Equ. Appl. 4 (2013), 43-57. MR 3064138 | Zbl 1330.35136 [3] M. L. Ahmed Oubeid, A. Benkirane, M. Sidi El Vally: Parabolic equations in Musielak-Orlicz-Sobolev spaces. Int. J. Anal. Appl. 4 (2014), 174-191. Zbl 06657910 [4] M. L. Ahmed Oubeid, A. Benkirane, M. Sidi El Vally: Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces. Bol. Soc. Paran. Mat. (3) 33 (2015), 193-223. MR 3267308 [5] M. Ait Khellou, A. Benkirane, S. M. Douiri: Existance of solutions for elliptic equations having naturel growth terms in Musielak-Orlicz spaces. J. Math. Comput. Sci. 4 (2014), 665-688. [6] E. Azroul, M. B. Benboubker, H. Redwane, C. Yazough: Renormalized solutions for a class of nonlinear parabolic equations without sign condition involving nonstandard growth. An. Univ. Craiova, Ser. Mat. Inf. 41 (2014), 69-87. MR 3234476 | Zbl 1324.35064 [7] E. Azroul, M. El Lekhlifi, H. Redwane, A. Touzani: Entropy solutions of nonlinear parabolic equations in Orlicz-Sobolev spaces, without sign condition and $L^1$ data. J. Nonlinear Evol. Equ. Appl. 2014 (2014), 101-130. MR 3322158 | Zbl 1327.35217 [8] E. Azroul, H. Hjiaj, A. Touzani: Existence and regularity of entropy solutions for strongly nonlinear $p(x)$-elliptic equations. Electron. J. Differ. Equ. 2013 (2013), Paper No. 68, 27 pages. MR 3040645 | Zbl 1291.35044 [9] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez: An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995), 241-273. MR 1354907 | Zbl 0866.35037 [10] A. Benkirane, M. Ould Mohamedhen Val: Some approximation properties in Musielak-Orlicz-Sobolev spaces. Thai. J. Math. 10 (2012), 371-381. MR 3001860 | Zbl 1264.46024 [11] A. Benkirane, M. Sidi El Vally: Variational inequalities in Musielak-Orlicz-Sobolev spaces. Bull. Belg. Math. Soc.-Simon Stevin 21 (2014), 787-811. MR 3298478 | Zbl 1326.35142 [12] M. S. B. Elemine Vall, A. Ahmed, A. Touzani, A. Benkirane: Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with $L^1$ data. Bol. Soc. Paran. Math. (3) 36 (2018), 125-150. DOI 10.5269/bspm.v36i1.29440 | MR 3632476 [13] A. Elmahi, D. Meskine: Parabolic equations in Orlicz spaces. J. Lond. Math. Soc., II. Ser. 72 (2005), 410-428. DOI 10.1112/S0024610705006630 | MR 2156661 | Zbl 1108.35082 [14] A. Elmahi, D. Meskine: Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 60 (2005), 1-35. DOI 10.1016/j.na.2004.08.018 | MR 2101516 | Zbl 1082.35085 [15] J. P. Gossez: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190 (1974), 163-205. DOI 10.2307/1996957 | MR 0342854 | Zbl 0239.35045 [16] S. Hadj Nassar, H. Moussa, M. Rhoudaf: Renormalized solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz spaces. Appl. Math. Comput. 249 (2014), 253-264. DOI 10.1016/j.amc.2014.10.026 | MR 3279419 | Zbl 1338.35257 [17] R. Landes, V. Mustonen: A strongly nonlinear parabolic initial boundary value problem. Ark. Mat. 25 (1987), 29-40. DOI 10.1007/BF02384435 | MR 0918378 | Zbl 0697.35071 [18] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Springer, Berlin (1983). DOI 10.1007/BFb0072210 | MR 0724434 | Zbl 0557.46020 [19] H. Redwane: Existence of a solution for a class of parabolic equations with three unbounded nonlinearities. Adv. Dyn. Syst. Appl. 2 (2007), 241-264. MR 2489045 [20] H. Redwane: Existence results for a class of nonlinear parabolic equations in Orlicz spaces. Electron. J. Qual. Theory Differ. Equ. 2010 (2010), Paper No. 2, 19 pages. DOI 10.14232/ejqtde.2010.1.2 | MR 2577155 | Zbl 1192.35103 [21] M. Sidi El Vally: Strongly nonlinear elliptic problems in Musielak-Orlicz-Sobolev spaces. Adv. Dyn. Syst. Appl. 8 (2013), 115-124. MR 3071548
Affiliations: Mohamed Saad Bouh Elemine Vall, Ahmed Ahmed, Abdelfattah Touzani, Abdelmoujib Benkirane, Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, B. P. 1796 Atlas, University of Sidi Mohamed Ibn Abdellah, 30003 Fez, Morocco e-mail: saad2012bouh@gmail.com, ahmedmath2001@gmail.com, atouzani07@gmail.com, abd.benkirane@gmail.com
