Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

TL;DR

This paper investigates conditions under which nonnegative weak solutions do not exist for certain quasilinear parabolic equations with potentials on complete noncompact Riemannian manifolds, emphasizing geometric and potential effects.

Contribution

It establishes nonexistence results linking manifold geometry, nonlinearity, and potential behavior at infinity for parabolic inequalities.

Findings

Nonexistence of solutions under specific geometric conditions

Influence of potential decay on solution existence

Interplay between manifold curvature and nonlinearity

Abstract

We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.