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

TL;DRAbstract

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 )<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 neighborh

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