On the notion of renormalized solution to nonlinear parabolic equations with general measure data

TL;DR

This paper introduces a new concept of renormalized solutions for nonlinear parabolic equations with general measure data, proving their existence and analyzing key properties in the context of possibly singular measures.

Contribution

It defines a novel notion of renormalized solutions for nonlinear parabolic equations with general measure data and establishes their existence.

Findings

Existence of renormalized solutions for equations with measure data.

Main properties of these solutions are characterized.

Applicable to equations with singular measure data.

Abstract

Here we introduce a new notion of renormalized solution to nonlinear parabolic problems with general measure data whose model is $$ \begin{cases} u_t-\Delta_{p} u =\mu & \text{in}\ (0,T)\times\Omega, u=u_0 & \text{on}\ \{0\} \times \Omega, u=0 &\text{on}\ (0,T)\times\partial\Omega, \end{cases} $$ for any, possibly singular, nonnegative bounded measure $\mu$. We prove existence of such a solutions and we discuss their main properties.