Nonexistence for hyperbolic problems on Riemannian manifolds1

Abstract

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined 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. 2010 Mathematics Subject Classification. 35B51,35B44, 35K08, 35K58, 35R01. Key words and phrases. Nonexistence of solutions, hyperbolic problems, wave equation. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni.

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