We consider the Cauchy problem for on the real line. It is shown that if , the solution has an additional logarithmic time decay in comparison with the free evolution in the sense of , . Moreover, the asymptotic profile of as 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.
Hideaki SUNAGAWA. "Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms." J. Math. Soc. Japan 58 (2) 379 - 400, April, 2006. https://doi.org/10.2969/jmsj/1149166781