Reduced measures associated to parabolic problems

TL;DR

This paper investigates the existence and properties of reduced measures for certain parabolic equations with boundary and initial conditions involving Radon measures, expanding understanding of measure solutions in parabolic PDEs.

Contribution

It introduces new results on the existence and characteristics of reduced measures for parabolic problems with measure data, generalizing previous theories.

Findings

Existence of reduced measures for specified parabolic equations.

Characterization of properties of these measures.

Extension of measure solution theory in parabolic PDEs.

Abstract

We study the existence and the properties of the reduced measures for the parabolic equations $\partial_tu-\Delta u+g(u)=0$ in $\Omega\times (0,\infty)$ subject to the conditions ($P$): $u=0$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=\mu$ and ($P'$): $u=\mu'$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=0$ where $\mu$ and $\mu'$ are positive Radon measures and $g$ a continuous nondecreasing function