Abstract
We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in \bm{R}
n× \bm{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-Δ u=|v|
p, \ ∂
t2v-Δ v=|u|
q in \bm{R}
n× \bm{R}, where (n+1)/(n-1)<p≤ q 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 Γ≤ 0, 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.
