Global existence and maximal regularity of solutions of gradient systems

In this article, we use a Galerkin method to prove a maximal regularity result for the following abstract gradient system{u′(t)+∇g(t)E(u(t))=f(t)for a.e.t∈(0,T),u(0)=u0. This abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.