Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 53, Number 4 (2001), 875-912.
Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions
We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in , where . As an application of the estimate, we study the asymptotic behavior as of solutions and to a system of semilinear wave equations: , in , where -1 with or . More precisely, it is known that there exists a critical curve on the p-q plane such that, when , the Cauchy problem for the system has a global solution with small initial data and that, when , a solution of the problem generically blows up in finite time even if the initial data are small. In this paper, when , we construct a global solution of the system which is asymptotic to a pair of solutions to the homogeneous wave equation with small initial data given, as , 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.
J. Math. Soc. Japan, Volume 53, Number 4 (2001), 875-912.
First available in Project Euclid: 29 May 2008
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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. https://projecteuclid.org/euclid.jmsj/1212067577