Instability and nonexistence of global solutions to nonlinear wave equations of the form
Author:
Howard A. Levine
Journal:
Trans. Amer. Math. Soc. 192 (1974), 121
MSC:
Primary 35L60; Secondary 47H15
MathSciNet review:
0344697
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Abstract: For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain , a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let be a continuously differentiable gradient operator with associated potential function . Assume that for all and some . Let where and be a solution to the equation in the title. The following statements hold: If , then for some . If and if u exists on , then (u,Pu) grows at least exponentially. If and and if the solution exists on , then (u,Pu) grows at least as fast as . A number of examples are given.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403446972
PII:
S 00029947(1974)03446972
Article copyright:
© Copyright 1974
American Mathematical Society
