Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions. (English) Zbl 1016.35050

Kubo, Hideo; Kubota, Kôji

J. Math. Soc. Japan 53, No. 4, 875-912 (2001).

The authors give a new a priori estimate for a classical solution of the nonhomogeneous wave equation in \(\mathbb{R}^n\times \mathbb{R},\) where \(n=2,3.\) Under some conditions on \(p, q,\) they construct a global solution of the system \(u_{tt}-\triangle u=|v|^p\), \(v_{tt}-\triangle v=|u|^q,\) which is asymptotic to a pair of solutions to a homogeneous wave equation with small initial data.

Reviewer: Marie Kopáčková (Praha)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B45	A priori estimates in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
35B33	Critical exponents in context of PDEs

Keywords:

semilinear wave equation; asymptotic behavior; Cauchy problem; coupled system of equations; critical estimates