Entropy solutions for nonlinear unilateral parabolic inequalities in Orlicz–Sobolev spaces
Volume 41 / 2014
Applicationes Mathematicae 41 (2014), 185-193
MSC: Primary 46E30; Secondary 35K85.
DOI: 10.4064/am41-2-6
Abstract
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 $L^1(Q)+W^{-1,x}E_{\overline {M}}(Q)$.
