## 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

#### 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 53, Number 4 (2001), 875-912.

**Dates**

First available in Project Euclid: 29 May 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1212067577

**Digital Object Identifier**

doi:10.2969/jmsj/05340875

**Mathematical Reviews number (MathSciNet)**

MR1852887

**Zentralblatt MATH identifier**

1016.35050

**Subjects**

Primary: 35L70: Nonlinear second-order hyperbolic equations 35B40: Asymptotic behavior of solutions 35B45: A priori estimates

**Keywords**

inhomogeneous wave equation semilinear wave equation scattering operator asymptotic behavior

#### Citation

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