Existence of renormalized solutions for a class of nonlinear parabolic equations with generalized growth in Orlicz spaces

Document Type : Original Article

Authors

Mohamed Bourahma

Abdelmoujib Benkirane

Jaouad Bennouna

Laboratory of mathematical analysis and applications (LAMA), Department of mathematics, Faculty of Sciences Dhar el Mahraz, Sidi Mohamed Ben Abdellah University, PB 1796 Fez-Atlas, Fez Morocco

10.22034/kjm.2020.184027.1422

Abstract

In this study, we prove an existence result of renormalized
solutions for nonlinear parabolic equations of the type
$$
\displaystyle\frac{\partial b(x,u)}{\partial t}
-\mbox{div}\>a(x,t,u,\nabla u)-\mbox{div}\>
\Phi(x,t,u)=
f \quad\mbox{in }{Q_T=\Omega\times (0,T)},
$$
where $b(x,\cdot)$ is a strictly increasing $C^1$-function for every $x\in\Omega$ with $b(x,0)=0$, the lower order term $\Phi$ satisfies a natural growth condition described by the appropriate Orlicz function $M$ and $f$ is an element of $L^1(Q_T)$. We don't assume any restriction neither on $M$ nor
on its conjugate $\overline{M}$.

Keywords

Parabolic problem

Orlicz spaces

Renormalized solutions

Generalized growth

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