On a class of nonlinear parabolic equations with natural growth in non-reflexive Musielak spaces

Document Type : Research Paper

Authors

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

2 Departement of Mathematics Faculte of Sciences SIdi Mohamed Ben Abdellah University Dhar Mahraz Fez Morocco

Abstract

An existence result of renormalized solutions for nonlinear parabolic Cauchy-Dirichlet problems whose model
$$\left\{\begin{array}{ll}
\displaystyle\frac{\partial b(x,u)}{\partial t}
-\mbox{div}\>\mathcal{A}(x,t,u,\nabla u)-\mbox{div}\>
\Phi(x,t,u)=
f &\mbox{ in }\Omega\times (0,T)\\
b(x,u)(t=0)=b(x,u_0) & \mbox{ in } \Omega\\
u=0 &\mbox{ on } \partial\Omega\times (0,T).
\end{array}\right.
$$
is given in the non reflexive Musielak spaces, 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 $\Phi$ is a non coercive Carath'{e}odory function satisfying only a natural growth condition described by the appropriate Musielak function $\varphi$ and $f$ is an integrable data.

Keywords