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Keywords:
inhomogeneous Musielak-Orlicz-Sobolev spaces; parabolic problems; Galerkin method
Summary:
We prove an existence result of entropy solutions for a class of strongly nonlinear parabolic problems in Musielak-Sobolev spaces, without using the sign condition on the nonlinearities and with measure data.
References:
[1] Aberqi, A., Bennouna, J., Mekkour, M., Redwane, H.: Existence results for a nonlinear parabolic problems with lower order terms. Int. J. Math. Anal. 7 (27) (2013), 1323–1340. DOI 10.12988/ijma.2013.13130 | MR 3053336
[2] Akdim, Y., Bennouna, J., Mekkour, M., Redwane, H.: Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities. Appl. Math. (Warsaw) 39 (1) (2012), 1–22. DOI 10.4064/am39-1-1 | MR 2880251
[3] Azroul, E., Benboubker, M.B., Redwane, H., Yazough, C.: Renormalized solutions for a class of nonlinear parabolic equations without sign condition involving nonstandard growth. An. Univ. Craiova Ser. Mat. Inform. 41 (1) (2014), 69–87. MR 3234476
[4] Azroul, E., Hjiaj, H., Touzani, A.: Existence and regularity of entropy solutions for strongly nonlinear $p(x)$-elliptic equations. EJDE 2013 (68) (2013), 1–27. MR 3040645
[5] Azroul, E., Lekhlifi, M. El, Redwane, H., Touzani, A.: Entropy solutions of nonlinear parablic equations in Orlicz–Sobolev spaces, without sign condition and $L^1$ data. J. Nonlinear Evolution Equa. and Appl. 2014 (7) (2014), 101–130. MR 3322158
[6] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (1995), 241–273. MR 1354907
[7] Benkirane, A., Sidi El Vally, M.: Some approximation properties in Musielak-Orlicz-Sobolev spaces. Thai. J. Math. 10 (2012), 371–381. MR 3001860
[8] Benkirane, A., Sidi El Vally, M.: Variational inequalities in Musielak-Orlicz-Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21 (5) (2014), 787–811. DOI 10.36045/bbms/1420071854 | MR 3298478
[9] Benkirane, A., Val), M. Sidi El Vally (Ould Mohamedhen: An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 20 (1) (2013), 57–75. DOI 10.36045/bbms/1366306714 | MR 3082743
[10] Boccardo, L., Dall’aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147 (1997), 237–258. DOI 10.1006/jfan.1996.3040 | MR 1453181
[11] Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure as data. J. Funct. Anal. 87 (1989), 149–169. DOI 10.1016/0022-1236(89)90005-0 | MR 1025884
[12] Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients. Nonlinear Anal. 19 (6) (1992), 581–597. DOI 10.1016/0362-546X(92)90023-8 | MR 1183665
[13] Elemine Vall, M.S.B., Ahmed, A., Touzani, A., Benkirane, A.: Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form. Math. Bohem. 143 (3) (2018), 225–249. DOI 10.21136/MB.2017.0087-16 | MR 3852293
[14] Elemine Vall, M.S.B., Ahmed, A., Touzani, A., Benkirane, A.: Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with $L^1$ data. Bol. Soc. Parana. Mat. (3) 36 (1) (2018), 125–150. DOI 10.5269/bspm.v36i1.29440 | MR 3632476
[15] Elmahi, A.: Strongly nonlinear parabolic initial-boundary value problems in Orlicz spaces. EJDE 09 (2002), 203–220, Proceedings of the 2002 Fez Conference on Partial Differential Equations. MR 1976696
[16] Gossez, J.-P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Amer. Math. Soc. 190 (1974), 163–205. DOI 10.1090/S0002-9947-1974-0342854-2 | MR 0342854
[17] Gwiazda, P., Minakowski, P., Wróblewska-Kamińska, A.: Elliptic problems in generalized Orlicz Musielak spaces. Cent. Eur. J. Math. 10 (6) (2012), 2019–2032. MR 2983144
[18] Gwiazda, P., Swierczewska-Gwiazda, A., Wróblewska-Kamińska, A.: Generalized Stokes system in Orlicz space. Discrete Contin. Dyn. Syst. 32 (6) (2012), 2125–2146. DOI 10.3934/dcds.2012.32.2125 | MR 2885802
[19] Gwiazda, P., Wittbold, P., Wróblewska-Kamińska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differential Equations 253 (2012), 635–666. DOI 10.1016/j.jde.2012.03.025 | MR 2921209
[20] Gwiazda, P., Wittbold, P., Wróblewska-Kamińska, A., Zimmermann, A.: Renormalized solutions to nonlinear parabolic problems in generalized Musielak Orlicz spaces. Nonlinear Anal. 129 (2015), 1–36. MR 3414919
[21] Khellou, M. Ait, Benkirane, A., Douiri, S.M.: Existence of solutions for elliptic equations having natural growth terms in Musielak Orlicz spaces. J. Math. Comput. Sci. 4 (4) (2014), 665–688.
[22] Khoi, L. Vy: On second order elliptic equations and variational inequalities with anisotropic principal operators. Topol. Methods Nonlinear Anal. 44 (1) (2014), 41–72. MR 3289007
[23] Landes, R., Mustonen, V.: A strongly nonlinear parabolic initial-boundary value problem. Ark. Math. 25 (1987), 29–40. DOI 10.1007/BF02384435 | MR 0918378
[24] Meskine, D.: Parabolic equations with measure data in Orlicz spaces. J. Evol. Equ. 5 (4) (2005), 529–543. DOI 10.1007/s00028-005-0217-8 | MR 2201525
[25] Musielak, J.: Modular spaces and Orlicz spaces. Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983, pp. iii+222. MR 0724434
[26] Oubeid, M.L. Ahmed, Benkirane, A., Vally, M. Sidi El: Nonlinear elliptic equations involving measure data in Musielak-Orlicz-Sobolev spaces. J. Abstr. Differ. Equ. Appl. 4 (2013), 43–57. MR 3064138
[27] Oubeid, M.L. Ahmed, Benkirane, A., Vally, M. Sidi El: Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces. Bol. Soc. Parana. Mat. (3) 33 (1) (2015), 193–225. MR 3267308
[28] Redwane, H.: Existence Results for a class of parabolic equations in Orlicz spaces. Electron. J. Qual. Theory Differ. Equ. 2010 (2) (2010), 1–19. DOI 10.14232/ejqtde.2010.1.2 | MR 2577155
[29] Simon, J.: Compact sets in the space $L^p(0, T, B)$. Ann. Mat. Pura Appl. 146 (1987), 65–96. DOI 10.1007/BF01762360 | MR 0916688
[30] Swierczewska, A., Gwiazda, P.: Nonlinear parabolic problems in Musielak Orlicz spaces. Nonlinear Anal. 98 (2014), 48–65. DOI 10.1016/j.na.2013.11.026 | MR 3158445
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