Nonexistence for hyperbolic problems on Riemannian manifolds

TL;DR

This paper derives necessary conditions for the existence of solutions to semilinear hyperbolic equations on Riemannian manifolds, extending known nonexistence results from Euclidean space to more general geometric settings.

Contribution

It generalizes nonexistence results for wave equations with power nonlinearities to complete noncompact Riemannian manifolds, incorporating spacetime-dependent weight functions.

Findings

Necessary conditions for solution existence are established.

Nonexistence results extend to manifolds beyond Euclidean space.

Inclusion of spacetime-dependent weight functions in the analysis.

Abstract

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.