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

Abstract

In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$\{\begin{array}{llc}{u}_t-{\Delta }_{p}u+|u{|}^{q-1}u=\lambda & \text{in}(0,T)\times \Omega u(0,x)=0& \text{in}\Omega ,\\ u(t,x)=0& \text{on}(0,T)\times \partial \Omega ,& \end{array}$$ where ${\Delta }_p=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ 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.