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October, 2001 Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions
J. Math. Soc. Japan 53(4): 875-912 (October, 2001). DOI: 10.2969/jmsj/05340875


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.


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Hideo KUBO. Kôji KUBOTA. "Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions." J. Math. Soc. Japan 53 (4) 875 - 912, October, 2001.


Published: October, 2001
First available in Project Euclid: 29 May 2008

zbMATH: 1016.35050
MathSciNet: MR1852887
Digital Object Identifier: 10.2969/jmsj/05340875

Primary: 35B40 , 35B45 , 35L70

Keywords: asymptotic behavior , inhomogeneous wave equation , scattering operator , semilinear wave equation

Rights: Copyright © 2001 Mathematical Society of Japan


Vol.53 • No. 4 • October, 2001
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