Nonexistence for hyperbolic problems on Riemannian manifolds¹

Article type: Research Article

Authors: Monticelli, Dario D.^a | Punzo, Fabio^a | Squassina, Marco^{b; *}

Affiliations: [a] Dipartimento di Matematica, Politecnico di Milano, Milano, Italy. E-mails: [email protected], [email protected] | [b] Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Note: [1] The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni.

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.

Keywords: Nonexistence of solutions, hyperbolic problems, wave equation

DOI: 10.3233/ASY-191580

Journal: Asymptotic Analysis, vol. 120, no. 1-2, pp. 87-101, 2020