Convergence properties of the strongly damped nonlinear Klein-Gordon equation

Abstract

We consider the nonlinear Klein-Gordon equation u tt + α(− Δ + γ)u i + (− Δ + m 2)u + λ ¦u¦ p − 1u = 0 over a domain Ω in R 3. In [2], Aviles and Sandefur established global existence of strong solutions when α > 0 for p > 3. For each α > 0 let u α be such a solution. In this paper we show that if Ω is a bounded domain with smooth boundary then there exists a sequence α k and a global weak solution v of the undamped equation (where α = 0) such that lim αk → 0 u αk = v in L 2(Ω) uniformly on any finite interval [0, T] with T > 0. We also show that if u is a strong local solution with α = 0, then for smooth enough initial data there exists an interval [0, T] such that lim α ↓ 0 u α = u uniformly in a much stronger norm. We conclude by noting a connection between these two results and by discussing the difficulty of extending the first result to the case Ω = R 3. We also note (in section 2) a natural extension of Theorem 1.1 to bounded domains in R n , n > 3.