Nonlinear parabolic problem with lower order terms in Musielak-Orlicz spaces

Volume 2, Issue 5, Page No 109-123, 2017

Author’s Name: Mhamed Elmassoudi^1,a), Ahmed Aberqi², Jaouad Bennouna¹

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¹University of Fez, Faculty of Sciences Dhar El Mahraz, Department of Mathematics, B.P 1796 Atlas Fez, Morocco.
²University of Fez, National School of Applied Sciences Fez,Morocco.

^a)Author to whom correspondence should be addressed. E-mail: elmassoudi09@gmail.com

Adv. Sci. Technol. Eng. Syst. J. 2(5), 109-123 (2017); DOI: 10.25046/aj020518

Keywords: Musielak-Orlicz space Nonlinear, Parabolic Problems, Entropy solution, Condition sign, Lower order term

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Abstract

We prove an existence result of entropy solutions for the nonlinear parabolic problems: $\frac{\partial b(x,u)}{\partial t} + A(u) - div(\Phi(x,t,u))+H(x,t,u,\nabla u) =f,$ and $A(u)=-div(a(x,t,u,\nabla u))$ is a Leary-Lions operator defined on the inhomogeneous Musielak-Orlicz space, the term $\Phi(x,t,u)$ is a Cratheodory function assumed to be continuous on u and satisfy only the growth condition $\Phi(x,t,u)\leq c(x,t)\overline{M}^{-1}M(x,\alpha_{0}u)$ , prescribed by Musielak-Orlicz functions $M$ and $\overline{M}$ which inhomogeneous and not satisfy $\Delta_2$ -condition, $H(x,t,u,\nabla u)$ is a Cratheodory function not satisfies neither the sign condition or coercivity and $f\in L^{1}(Q_T)$ .

Received: 05 May 2017, Accepted: 15 July 2017, Published Online: 28 December 2017

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Nonlinear parabolic problem with lower order terms in Musielak-Orlicz spaces