Diffuse measures and nonlinear parabolic equations

Abstract

Given a parabolic cylinder Q = (0,T) ! ! , where ! R N is a bounded domain, we prove new properties of solutions of ut # pu = µ in Q with Dirichlet boundary conditions, where µ 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., mea- sures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures µk = (Tk(u))t# p(Tk(u)),andweintroduceanewnotionofrenormalizedsolutionbasedonthisproperty. We finally apply our new approach to prove the existence of solutions of ut # pu + h(u) = µ in Q, for any function h such that h(s)s $ 0 and for any diffuse measure µ; 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.