Open AccessArticle
Nonlinear parabolic problem with lower order terms in Musielak-Orlicz spaces
1 University of Fez, Faculty of Sciences Dhar El Mahraz, Department of Mathematics, B.P 1796 Atlas Fez, Morocco.
2 University of Fez, National School of Applied Sciences Fez,Morocco.
*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
Received: 5 May 2017, Accepted: 15 July 2017, Published Online: 28 December 2017
(This article belongs to Section Applied Mathematics (MAP))
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 \(f\in L^{1}(Q_T)\).
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