Nonlinear Analysis 72 (2010) 2658–2683 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms Mohammad A. Rammaha a,∗ , Sawanya Sakuntasathien b a Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA b Mathematics Department, Faculty of Science, Silpakorn University, Nakhonpathom, Thailand article info Article history: Received 12 August 2009 Accepted 3 November 2009 MSC: primary 35L05 35L20 secondary 58G16 Keywords: Wave equations Damping and source terms Weak solutions Blow up of solutions Energy identity abstract We focus on the global well-posedness of the system of nonlinear wave equations u tt − ∆u + (d|u| k + e|v| l )|u t | m−1 u t = f 1 (u,v) v tt − ∆v + (d ′ |v| θ + e ′ |u| ρ )|v t | r −1 v t = f 2 (u,v), in a bounded domain Ω ⊂ R n , n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f 1 (u,v) and f 2 (u,v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. The model Many questions in physics and engineering give rise to problems that deal with coupled evolution equations. For instance, in scattering theory and certain mechanical applications, such evolution equations come in the form of a system of nonlinear wave equations. An important example of such systems goes back to Reed [1] in 1976 who proposed a system in three space dimensions which is similar to system (1.1), but without the presence of any damping. In this article, we study a system of nonlinear wave equations which features two competing forces. One force is a degenerate damping term and the other is a strong source. In particular, we analyze the influence of these forces on the long-time behavior of solutions. Let F : R 2 −→ R be the C 1 -function given by F (u,v) = a |u + v| p+1 + 2b |uv| p+1 2 , where p ≥ 3, a > 1 and b > 0. Throughout the paper, Ω is a bounded open connected set in R n , n = 1, 2, 3, with a smooth boundary Γ = ∂ Ω. We study the global well-posedness of the following initial boundary value problem: u tt − ∆u + (d|u| k + e|v| l )|u t | m−1 u t = f 1 (u,v), in Ω × (0, T ) ≡ Q T , ∗ Corresponding author. Tel.: +1 402 472 7258; fax: +1 402 472 8466. E-mail addresses: rammaha@math.unl.edu, mrammaha1@math.unl.edu (M.A. Rammaha), ssawanya@hotmail.com (S. Sakuntasathien). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.11.013