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

Document Type : Original Article

Authors

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