Diffuse measures and nonlinear parabolic equations

2025/08/08 by Petitta, Francesco, Ponce, Augusto C., Porretta, Alessio · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.06384

Abstract

Given a parabolic cylinder Q =(0,T)×Ω, where Ω⊂ ℝ^N is a bounded domain, we prove new properties of solutions of u_t-Δ_p u = 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.\@ 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-Δ_p u + 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.

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• Regularity and existence for a mixed local-nonlocal parabolic equation with variable singularities and measure data

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