Regularity for entropy solutions of parabolic P-Laplacian equations

In this note we give some summability results for entropy solu-tions of the nonlinear parabolic equation ut − div ap(x,∇u) = f in]0, T [×Ω with initial datum in L1(Ω) and assuming Dirichlet’s boundary condition, where ap(.,.) is a Carathéodory function sat-isfying the classical Leray-Lions hypotheses, f ∈ L1(]0, T [×Ω) and Ω is a domain inRN. We find spaces of type Lr(0, T;Mq(Ω)) con-taining the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian equation is considered. 1