This article is cited in 8 papers

Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity

M. O. Korpusov, A. A. Panin

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: We consider an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Under certain conditions on the operators in the equation, we prove its local (in time) solubility and give sufficient conditions for finite-time blow-up of solutions of the corresponding abstract Cauchy problem. The proof uses a modification of a method of Levine. We give examples of Cauchy problems and initial-boundary value problems for concrete non-linear equations of mathematical physics.

Keywords: finite-time blow-up, generalized Klein–Gordon equations, non-linear hyperbolic equations, non-linear mixed boundary-value problems, field theory.

UDC: 517.957

MSC: 35B44, 35L20, 35L70

Received: 17.03.2013

DOI: 10.4213/im8118

 English version: Izvestiya: Mathematics, 2014, 78:5, 937–985

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