On the Asymptotic Behavior of Nonlinear Wave Equations

Positive energy solutions of the Cauchy problem for the equation <!-- MATH $\square u = {m^2}u + F(u)$ --> are considered. With <!-- MATH $G(u) = \smallint _0^uF(s)ds$ --> , it is proven that must be nonnegative in order for uniform decay and the existence of asymptotic ``free'' solutions to hold. When 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 has the form <!-- MATH $|u{|^{p - 1}}u$ --> , scattering theory is shown to be impossible if <!-- MATH $1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$ --> <img width="226" height="43" align="MIDDLE" border="0" src="images/img8.gif" alt="$ 1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$">.