On the asymptotic behavior of nonlinear wave equations

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 $1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$.