Diffuse measures and nonlinear parabolic equations

Abstract

Given a parabolic cylinder $Q =(0,T)\times\Omega$, 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 $\mu$ 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 $$ u_t-\Delta_{p} u + h(u)=\mu \quad \text{in $Q$,} $$ for any function $h$ such that $h(s)s\geq 0$ 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.