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

Abstract

We consider the Cauchy problem for ∂_t²u-∂_x²u+u = -g(∂_tu)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≤p≤∞. Moreover, the asymptotic profile of u(t,x) as t→+∞ 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.