Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms

Abstract

We consider the Cauchy problem for ${\partial }_t^{2}u-{\partial }_x^{2}u+u=-g({\partial }_{t}u{)}^3$ on the real line. It is shown that if $g>0$, the solution has an additional logarithmic time decay in comparison with the free evolution in the sense of $L^p$, $2\le p\le \mathrm{\infty }$. Moreover, the asymptotic profile of $u(t,x)$ as $t\to +\mathrm{\infty }$ is obtained. We also discuss a generalization. Consequently we see that the "null condition" in the sense of J.-M. Delort (Ann. Sci. École Norm. Sup., 34 (2001), 1--61) is not optimal for small data global existence for nonlinear Klein-Gordon equations.