Existence of renormalized solution for a class of doubly nonlinear parabolic equation with nonstandard growth

Abstract

We prove the existence of a renormalized solution to a class of doubly nonlinear parabolic equation
\begin{equation*}
\dfrac{\partial b(x,u)}{\partial t} -\text{div}(a(x,t,u,\nabla u)+\Phi (u)) =f- \text{ div}(F) \quad \mbox{in} \quad Q,
\end{equation*}
where $-\text{div}a(x,t,u,\nabla u)$ is a Leray-Lions operator which is coercive and which grows like $|\nabla u|^{p(x)-1}$
with respect to $\nabla u$, but which is not
restricted by any growth condition with respect to $u$ and where $b(x,u)$ is an $C^1(\mathbb{R})$-function strictly increasing with respect $u$.
The data $f$, $F$ and $u_0$ respectively belong to $L^1(Q)$, $ (L^{p^\prime(\cdot)}(Q))^{N}$ and $L^1(\Omega)$.