Diffuse measures and nonlinear parabolic equations

Given a parabolic cylinder Q = (0, T) x Omega, where Omega subset of R(N) is a bounded domain, we prove new properties of solutions of u(t) - Delta pu = mu in Q with Dirichlet boundary conditions, where mu is a finite Radon measure in Q. We first prove a priori estimates on the p-parabolic capacity of level sets of u. We then show that diffuse measures (i.e., measures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures mu(k) = (T(k) (u))(t)-Delta(p)(T(k) (u)), and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of u(t) - Delta pu + h(u) = mu in Q, for any function h such that h(s) s >= 0 and for any diffuse measure mu; when h is nondecreasing, we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.