International Journal of Nonlinear Analysis and Applications

On a class of nonlinear parabolic equations with natural growth in non-reflexive Musielak spaces

Document Type : Research Paper

Authors

1 Laboratory of mathematical analysis and applications (LAMA), Department of mathematics, Faculty of Sciences Dhar el Mahraz, Sidi Mohamed Ben Abdellah University, PB 1796 Fez, Morocco

2 Departement of Mathematics Faculte of Sciences SIdi Mohamed Ben Abdellah University Dhar Mahraz Fez Morocco

Abstract

An existence result of renormalized solutions for nonlinear parabolic Cauchy-Dirichlet problems whose model
$$\left\{\begin{array}{ll}
\displaystyle\frac{\partial b(x,u)}{\partial t}
-\mbox{div}\>\mathcal{A}(x,t,u,\nabla u)-\mbox{div}\>
\Phi(x,t,u)=
f &\mbox{ in }\Omega\times (0,T)\\
b(x,u)(t=0)=b(x,u_0) & \mbox{ in } \Omega\\
u=0 &\mbox{ on } \partial\Omega\times (0,T).
\end{array}\right.
$$
is given in the non reflexive Musielak spaces, where $b(x,\cdot)$ is a strictly increasing $C^1$-function for every $x\in\Omega$ with $b(x,0)=0$, the lower order term $\Phi$ is a non coercive Carath'{e}odory function satisfying only a natural growth condition described by the appropriate Musielak function $\varphi$ and $f$ is an integrable data.

Keywords

[1] A. Aberqi, J. Bennouna and H. Redwane, A nonlinear parabolic problems with lower order terms and measure data, Thai J. Math. 14(1) (2016) 115–130.
[2] A. Aberqi, J. Bennouna and M. Elmassoudi, Existence and uniqueness of renormalized solution for nonlinear parabolic equations in Musielak-Orlicz spaces, Bol. Soc. Paran. Mat. (2019) doi:10.5269/bspm.45234.
[3] R. Adams, Sobolev Spaces, Academic Press Inc, New York. 1975.
[4] M.L. Ahmed Oubeid, A. Benkirane and M. Sidi El Vally, Strongly nonlinear parabolic problems in Musielak-Orlicz Sobolev spaces, Bol. Soc. Paran. Mat. (2015) 191–223.
[5] M.L. Ahmed Oubeid, A. Benkirane and M. Sidi El Vally, Parabolic equations in Musielak-Orlicz-Sobolev spaces, Int. J. Anal. Appl. 4(2) (2014) 174–191.
[6] Y. Ahmida, I. Chlebicka, P. Gwiazda and A. Youssfi, Gossez’s approximation theorems in Musielak-Orlicz-Sobolev spaces, J. Funct. Anal. (2018), https://doi.org/10.1016/j.jfa.2018.05.015
[7] M. Ait Khellou and A. Benkirane, Elliptic inequalities with L1 data in Musielak-Orlicz spaces, Monat. Math. 183 (2017) 33.
[8] M. Ait Khellou and A. Benkirane, Correction to elliptic inequalities with L1 data in Musielak-Orlicz spaces, Monat. Math. 187 (2018) 181–187.
[9] M. Ait Khellou, A. Benkirane and S. M. Douiri, Existence of solutions for elliptic equations having natural growth terms in Musielak-Orlicz spaces, J. Math. Comput. Sci. 4(4) (2014) 665–688.
[10] M. Ait Khellou and A. Benkirane, Renormalized solution for nonlinear elliptic problems with lower order terms and L1 data in Musielak-Orlicz spaces, Ann. University of Craiova, Math. Comput. Sci. Ser. 43(2) (2016)164–187.
[11] A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl., IV. Ser. 182(1) (2003) 53–79.
[12] A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of non uniformly elliptic equations, Atti Semin. Mat. Fis. Univ. Modena 46-suppl., (1998) 381–391.
[13] E. Azroul, H. Redwane and M. Rhoudaf, Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces, Port. Math. 66(1) (2009) 29–63.
[14] A. Benkirane and M. Sidi El Vally (Ould Mohameden Val), Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math.Soc. Simon Stevin, 21(5) (2014) 787–811.
[15] A. Benkirane and M. Sidi El Vally (Ould Mohameden Val), An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math.Soc. Simon Stevin 20 (2013) 57–75.
[16] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston. 1988.
[17] J. Bennouna, M. Hammoumi and A. Aberqi, Nonlinear degenerated parabolic equations with lower order terms, Elec. J Math. Anal. Appl. (2016) 234–253.
[18] D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with L1 data, existence and uniqueness, Proc. R. Soc. Edinburgh Sect. A 127 (1997) 1137–1152.
[19] D. Blanchard, F. Murat and H. Redwane, Existence et unicite de la solution renormalisee dun probleme parabolique assez general, C. R. Acad. Sci. Paris Ser. 1329 (1999) 575–580.
[20] D. Blanchard and A. Porretta, A Stefan problems with diffusion and convection, Differ. Equ. 210 (2005) 383–428.
[21] L. Boccardo, A. Dall’Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Semin. Mat. Fis. Univ. Modena 46-suppl. (1998) 51–81.
[22] M. Bourahma, A. Benkirane and J. Bennouna, Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces, J. Rend. Circ. Mat. Palermo, II. Ser (2019) https://doi.org/10.1007/s12215-019-00399-z
[23] M. Bourahma, A. Benkirane and J. Bennouna, An existence result of entropy solutions to elliptic problems in generalized Orlicz-Sobolev spaces, J. Rend. Circ. Mat. Palermo, II. Ser (2020) https://doi.org/10.1007/s12215-020-00506-5
[24] M. Bourahma, J. Bennouna and M. El Moumni, Existence of a weak bounded solutions for nonlinear degenerate elliptic equations in Musielak spaces, Moroccan J. of Pure and Appl. Anal. (MJPAA) 6(1) (2020) 16–33.
[25] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solution of elliptic equation with general measure data. Ann. Scuola Norm. Sup. Pisa CI. Sci. 28(4) (1999) 741–808.
[26] M.S.B. Elemine Vall, A. Ahmed, A. Touzani and A. Benkirane, Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with L1 data, Bol. Soc. Paran. Mat. (3s.)  36(1) (2018) 125–150.
[27] B. El Haji, M. El Moumni and K. Kouhaila, On nonlinear elliptic problems having large monotonicity with L1-data in weighted Orlicz-Sobolev spaces, Moroccan J. Pure Appl. Anal. 5 (1) 104–116.
[28] A. Elmahi and D. Meskine, Strongly nonlinear parabolic equations with natural growth terms and L1 data in Orliczspaces, Port. Math. Nova 62 (2005) 143–183.
[29] A. Elmahi and D. Meskine, Strongly nonlinear parabolic equations with natural growth terms and L1 data inOrlicz spaces. Portugaliae Mathematica. Nova 62 (2005) 143–183.
[30] J.P. Gossez, A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc. Sympos. Pure Math. 45 (1986) 455 462.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 1207-1233
  • Receive Date: 19 August 2020
  • Revise Date: 08 January 2021
  • Accept Date: 06 March 2021