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Transactions of the American Mathematical Society

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On the asymptotic behavior of nonlinear wave equations


Author: Robert T. Glassey
Journal: Trans. Amer. Math. Soc. 182 (1973), 187-200
MSC: Primary 35L05
DOI: https://doi.org/10.1090/S0002-9947-1973-0330782-7
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 $ \vert u{\vert^{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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DOI: https://doi.org/10.1090/S0002-9947-1973-0330782-7
Keywords: Positive energy solutions, absence of uniform decay, non-existence of scattering
Article copyright: © Copyright 1973 American Mathematical Society

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