Journal of the Mathematical Society of Japan

Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions

Hideo KUBO and Kôji KUBOTA

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We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in Rn×R, where n=2,3. As an application of the estimate, we study the asymptotic behavior as t± of solutions u(x,t) and v(x,t) to a system of semilinear wave equations: t2u-Δu=|v|p, t2v-Δv=|u|q in Rn×R, where (n+1)/(n -1 )<pq with n=2 or n=3. More precisely, it is known that there exists a critical curve Γ=Γ(p,q,n)=0 on the p-q plane such that, when Γ>0, the Cauchy problem for the system has a global solution with small initial data and that, when Γ0, a solution of the problem generically blows up in finite time even if the initial data are small. In this paper, when Γ>0, we construct a global solution (u(x,t),v(x,t)) of the system which is asymptotic to a pair of solutions to the homogeneous wave equation with small initial data given, as t-, in the sense of both the energy norm and the pointwise convergence. We also show that the scattering operator exists on a dense set of a neighborhood of 0 in the energy space.

Article information

J. Math. Soc. Japan, Volume 53, Number 4 (2001), 875-912.

First available in Project Euclid: 29 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations 35B40: Asymptotic behavior of solutions 35B45: A priori estimates

inhomogeneous wave equation semilinear wave equation scattering operator asymptotic behavior


KUBO, Hideo; KUBOTA, Kôji. Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions. J. Math. Soc. Japan 53 (2001), no. 4, 875--912. doi:10.2969/jmsj/05340875.

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