Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains

TL;DR

This paper investigates conditions under which solutions do not exist for certain quasilinear parabolic equations with potentials in bounded domains, focusing on boundary behavior and nonlinearities, and establishing sharp growth criteria for nonexistence.

Contribution

It provides new nonexistence results for quasilinear parabolic equations with potentials, especially characterizing the critical boundary growth rate in the semilinear case.

Findings

Nonexistence results depend on potential behavior near the boundary.

Sharp critical growth rate of the potential ensures nonexistence in semilinear cases.

Boundary behavior of the potential is crucial for solution existence.

Abstract

We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $\Omega\times(0,+\infty)$, where $\Omega$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.