Decay of solutions of nonlinear wave equations in three space dimensions

Given a solution of the Cauchy problem for nonlinear wave equations of the type $\text{?}{}^2\text{u}\text{?t}{}^2\text{? Δu + f(u) = 0}$ in three space dimensions the asymptotic behaviour in time is considered. It is shown that for nonlinearities which behave like powers $\text{¦u¦}{}^{\mathrm{\sigma \; ?\; 1}}\text{u}$ uniform decay holds with a certain rate depending on σ if $\text{5 > σ >}\text{1}\text{2}\text{+}\text{1}\text{2}\text{√13}$, and moreover scattering states exist if σ is not too small. This improves former results of W. A. Strauss (J. Funct. Anal. 2 (1968), 409–457).