Instability and nonexistence of global solutions to nonlinear wave equations of the form

Author:
Howard A. Levine

Journal:
Trans. Amer. Math. Soc. **192** (1974), 1-21

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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DOI:
https://doi.org/10.1090/S0002-9947-1974-0344697-2

Article copyright:
© Copyright 1974
American Mathematical Society