Entropy solutions for nonlinear unilateral parabolic inequalities in Orlicz–Sobolev spaces

Azeddine Aissaoui Fqayeh¹ ; Abdelmoujib Benkirane¹ ; Mostafa El Moumni¹

¹ Laboratory LAMA Department of Mathematics Faculty of Sciences Dhar El Mahraz University Sidi Mohamed Ben Abdellah P.O. Box 1796 Atlas Fez, Morocco

Applicationes Mathematicae, Tome 41 (2014) no. 2-3, pp. 185-193.

Résumé

We discuss the existence of entropy solution for the strongly nonlinear unilateral parabolic inequalities associated to the nonlinear parabolic equations $\frac{\partial u}{\partial t}-\mathrm{d}\mathrm{i}\mathrm{v}(a(x,t,u,\mathrm{\nabla }u)+\Phi (u))+g(u)M(|\mathrm{\nabla }u|)=\mu$ in $Q,$ in the framework of Orlicz–Sobolev spaces without any restriction on the $N$-function of the Orlicz spaces, where $-\mathrm{d}\mathrm{i}\mathrm{v}(a(x,t,u,\mathrm{\nabla }u))$ is a Leray–Lions operator and $\Phi \in C^0(\mathbb{R},{\mathbb{R}}^N)$. The function $g(u)M(|\mathrm{\nabla }u|)$ is a nonlinear lower order term with natural growth with respect to $|\mathrm{\nabla }u|$, without satisfying the sign condition, and the datum $\mu$ belongs to $L^1(Q)$ or $L^1(Q)+W^{-1,x}{E}_{\bar{M}}(Q)$.

Détail

DOI : 10.4064/am41-2-6

Mots-clés : discuss existence entropy solution strongly nonlinear unilateral parabolic inequalities associated nonlinear parabolic equations frac partial partial div u nabla varphi nabla framework orlicz sobolev spaces without restriction n function orlicz spaces where div u nabla leray lions operator varphi mathbb mathbb function nabla nonlinear lower order term natural growth respect nabla without satisfying sign condition datum belongs overline