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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the asymptotic behavior of nonlinear wave equations

Author: Robert T. Glassey
Journal: Trans. Amer. Math. Soc. 182 (1973), 187-200
MSC: Primary 35L05
MathSciNet review: 0330782
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Abstract: Positive energy solutions of the Cauchy problem for the equation $\square u = {m^2}u + F(u)$ are considered. With $G(u) = \smallint _0^uF(s)ds$, it is proven that $G(u)$ must be nonnegative in order for uniform decay and the existence of asymptotic “free” solutions to hold. When $G(u)$ is nonnegative and satisfies a growth restriction at infinity, the kinetic and potential energies (with m = 0) are shown to be asymptotically equal. In case $F(u)$ has the form $|u{|^{p - 1}}u$, scattering theory is shown to be impossible if $1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$.

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Keywords: Positive energy solutions, absence of uniform decay, non-existence of scattering
Article copyright: © Copyright 1973 American Mathematical Society