Convergence properties of the strongly damped nonlinear Klein-Gordon equation

AbstractWe consider the nonlinear Klein-Gordon equation utt + α(− Δ + γ)ui + (− Δ + m2)u + λ ¦u¦p − 1u = 0 over a domain Ω in R3. In [2], Aviles and Sandefur established global existence of strong solutions when α > 0 for p > 3. For each α > 0 let uα be such a solution. In this paper we show that if Ω is a bounded domain with smooth boundary then there exists a sequence αk and a global weak solution v of the undamped equation (where α = 0) such that limαk → 0uαk = v in L2(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 limα ↓ 0 uα = 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 = R3. We also note (in section 2) a natural extension of Theorem 1.1 to bounded domains in Rn, n > 3