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arXiv · 1506.04567

About Blow up of Solutions With Arbitrary Positive Initial Energy to Nonlinear Wave Equations

B. A. Bilgin·V. K. Kalantarov
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Abstract

We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem for the abstract wacve eqation of the form $Pu_{tt}+Au=F(u) \ (*)$ in a Hilbert space, where $P,A$ are positive linear operators and $F(\cdot)$ is a continuously differentiable gradient operator can be obtained from the result of H.A. Levine on the growth of solutions of the Cauchy problem for (*). This result is applied to the study of inital boundary value problems for nonlinear Klein-Gordon equations, generalized Boussinesq equations and nonlinear plate equations. A result on blow up of solutions with positive initial energy of the initial boundary value problem for wave equation under nonlinear boundary condition is also obtained.

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B. A. Bilgin, V. K. Kalantarov. 2015-06-15. About Blow up of Solutions With Arbitrary Positive Initial Energy to Nonlinear Wave Equations. https://arxiv.org/abs/1506.04567

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