Global smooth solutions for a non-linear system of wave equations

In this paper we consider the Cauchy problem for the non-linear system of wave equations with Hamilton structure {utt−Δu=−F1(|u|2,|v|2)u,vtt−Δv=−F2(|u|2,|v|2)v where there exists a function F(λ,μ)F(λ,μ) such that ∂F(λ,μ)∂λ=F1(λ,μ),∂F(λ,μ)∂μ=F2(λ,μ).On the basis of a Morawetz–Pohožaev dilation identity derived for the system, we prove that potential energy cannot concentrate at any fixed point; combining this with the improved time–space estimate presented in [J. Shatah, M. Struwe, Regularity results for non-linear wave equations, Ann. of Math. 138 (1993) 503–518], we obtain global smooth solutions of the system and, by the energy method, we prove that those solutions are of class C∞C∞ if the non-linearities and initial data are smooth enough.