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

Abstract

We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in $R^n\times R$, where $n=2,3$. As an application of the estimate, we study the asymptotic behavior as $t\to \pm \infty$ of solutions $u(x,t)$ and $v(x,t)$ to a system of semilinear wave equations: ${\partial }_t^{2}u-\Delta u=|v{|}^p$, ${\partial }_t^{2}v-\Delta v=|u{|}^q$ in $R^n\times R$, where $(n+1)/(n$ -1 $)$ with $n=2$ or $n=3$. More precisely, it is known that there exists a critical curve $\Gamma =\Gamma (p,q,n)=0$ on the p-q plane such that, when $\Gamma >0$, the Cauchy problem for the system has a global solution with small initial data and that, when $\Gamma \le 0$, a solution of the problem generically blows up in finite time even if the initial data are small. In this paper, when $\Gamma >0$, we construct a global solution $\left(u\right(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 $t\to -\infty$, 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.