A nonexistence result for nonlinear parabolic equations with singular measures as data

TL;DR

This paper establishes a nonexistence result for certain nonlinear parabolic equations with measure data concentrated on sets of zero parabolic capacity, highlighting limitations in solution existence under these conditions.

Contribution

It proves that solutions do not exist for nonlinear parabolic equations with measure data supported on zero capacity sets when the nonlinearity exponent is sufficiently large.

Findings

Nonexistence of solutions for measure data on zero capacity sets

Conditions under which solutions cannot be obtained

Implications for the theory of nonlinear parabolic equations

Abstract

In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- \Delta_p u+|u|^{q-1}u=\lambda & \text{in}\ (0,T)\times\Omega u(0,x)=0 & \text{in}\ \Omega,\\ u(t,x)=0 & \text{on}\ (0,T)\times\partial\Omega, \end{cases} $$ where $\Delta_p ={\rm div }(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-laplace operator, $\lambda$ is measure concentrated on a set of zero parabolic $r$-capacity ($1<p<r$), and $q$ is large enough.