## Journal of the Mathematical Society of Japan

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

Hideaki SUNAGAWA

#### 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 $L^p$, $2\leq p \leq \infty$. Moreover, the asymptotic profile of $u(t,x)$ as $t \to +\infty$ 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 58, Number 2 (2006), 379-400.

Dates
First available in Project Euclid: 1 June 2006

https://projecteuclid.org/euclid.jmsj/1149166781

Digital Object Identifier
doi:10.2969/jmsj/1149166781

Mathematical Reviews number (MathSciNet)
MR2228565

Zentralblatt MATH identifier
1107.35087

#### Citation

SUNAGAWA, Hideaki. Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms. J. Math. Soc. Japan 58 (2006), no. 2, 379--400. doi:10.2969/jmsj/1149166781. https://projecteuclid.org/euclid.jmsj/1149166781

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