Regularity for entropy solutions of parabolic $p$-Laplacian type equations

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation $u_t -\operatorname{div} {\mathbf a}_p (x,\nabla u) = f$ in $]0,T[\times \Omega$ with initial datum in $L^1(\Omega)$ and assuming Dirichlet's boundary condition, where ${\mathbf a}_p(.,.)$ is a Carathéodory function satisfying the classical Leray-Lions hypotheses, $f\in L^1(]0,T[\times \Omega)$ and $\Omega$ is a domain in ${\mathbb R}^N$. We find spaces of type $L^r(0,T; {\cal M}^q(\Omega))$ containing the entropy solution and its gradient. We also include some summability results when $f = 0$ and the $p$-Laplacian equation is considered.