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}-{\rm div}(a(x,t,u,\nabla u)+\varPhi (u)) + g(u)M(|\nabla u|) = \mu $ in $Q,$ in the framework of Orlicz–Sobolev spaces without any restriction on the $N$-function of the Orlicz spaces, where $-{\rm div} (a(x,t,u,\nabla u))$ is a Leray–Lions operator and $\varPhi \in C^{0}(\mathbb {R},\mathbb {R}^{N})$. The function $g(u)M(|\nabla u|)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, without satisfying the sign condition, and the datum $\mu $ belongs to $L^1(Q)$ or $L^1(Q)+W^{-1,x}E_{\overline {M}}(Q)$.

Détail

DOI : 10.4064/am41-2-6

Keywords: 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