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.
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. https://doi.org/10.2969/jmsj/05340875