Diffuse measures and nonlinear parabolic equations

Abstract

Given a parabolic cylinder Q = (0, T) × Ω, where ${\Omega\subset \mathbb {R}^N}$ is a bounded domain, we prove new properties of solutions of

$$u_t-\Delta_p u = \mu \quad \text{in }Q$$

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., measures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures μ _k = (T _k (u)) _t −Δ _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

$$u_t-\Delta_{p} u + h(u)=\mu \quad \text{in }Q,$$

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.