Journal of the Mathematical Society of Japan

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

Hideaki SUNAGAWA

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Abstract

We consider the Cauchy problem for t 2 u - x 2 u + u = - g ( 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 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.

Article information

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

Dates
First available in Project Euclid: 1 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1149166781

Digital Object Identifier
doi:10.2969/jmsj/1149166781

Mathematical Reviews number (MathSciNet)
MR2228565

Zentralblatt MATH identifier
1107.35087

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35L15: Initial value problems for second-order hyperbolic equations

Keywords
Klein-Gordon equation nonlinear dissipation large time asymptotics null condition

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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References

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