On the asymptotic behavior of nonlinear wave equations

Abstract

Positive energy solutions of the Cauchy problem for the equation ◻ u = m 2 u + F ( u ) \square u = {m^2}u + F(u) are considered. With G ( u ) = ∫ 0 u F ( s ) d s G(u) = \smallint _0^uF(s)ds , it is proven that G ( u ) G(u) must be nonnegative in order for uniform decay and the existence of asymptotic “free” solutions to hold. When G ( u ) 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 ) F(u) has the form | u | p − 1 u |u{|^{p - 1}}u , scattering theory is shown to be impossible if 1 > p ≤ 1 + 2 n − 1 ( n ≥ 2 ) 1 > p \leq 1 + 2{n^{ - 1}}\;(n \geq 2) .