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Global existence, decay and blow up solutions for coupled nonlinear wave equations with damping and source terms

We study the initial-boundary value problem for a system of nonlinear wave equations with nonlinear damping and source terms, in a bounded domain. The decay estimates of the energy function are established by using Nakao’s inequality. The nonexistence of global solutions is discussed under some conditions on the given parameters. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 633 – 651 ¨ ITAK ˙ c TUB doi:10.3906/mat-1110-48 Global existence, decay and blow up solutions for coupled nonlinear wave equations with damping and source terms ˙ ¸ KIN, ˙ ∗ Necat POLAT Erhan PIS Department of Mathematics, Dicle University, 21280, Diyarbakır, Turkey Received: 24.10.2011 • Accepted: 17.04.2012 • Published Online: 12.06.2013 • Printed: 08.07.2013 Abstract: We study the initial-boundary value problem for a system of nonlinear wave equations with nonlinear damping and source terms, in a bounded domain. The decay estimates of the energy function are established by using Nakao’s inequality. The nonexistence of global solutions is discussed under some conditions on the given parameters. Key words: Decay rate, blow up, initial boundary value problem, nonlinear wave equations 1. Introduction In this paper we consider the following initial-boundary value problem: ⎧ m−1 2 ⎪ ∇u + f1 (u, v) , (x, t) ∈ Ω × (0, T ) , + |u | u = div ρ |∇u| u ⎪ tt t t ⎪ ⎪ ⎪ ⎪ r−1 2 ⎨ vtt + |vt | vt = div ρ |∇v| ∇v + f2 (u, v) , (x, t) ∈ Ω × (0, T ) , u = v = 0, (x, t) ∈ ∂Ω × (0, T ) , ⎪ ⎪ ⎪ ⎪ (x) , u (x, 0) = u (x) , x ∈ Ω, u (x, 0) = u ⎪ 0 t 1 ⎪ ⎩ v (x, 0) = v0 (x) , vt (x, 0) = v1 (x) , x ∈ Ω, (1.1) where Ω is a bounded domain with smooth boundary ∂Ω in Rn , n = 1, 2, 3; m, r ≥ 1; fi (. , .) : R2 −→ R are given functions to be specified later. Problems of this type arise in material science and physics. We assume that ρ is a function which satisfies the relation ρ (s) ∈ C 1 , ρ (s) > 0, ρ (s) + 2sρ (s) > 0 (1.2) for s > 0. (A1). Let F (u, v) = a |u + v| n = 3; f1 (u, v) = ∂F ∂u , f2 (u, v) = ∂F ∂v p+1 + 2b |uv| p+1 2 with a, b > 0, p ≥ 3 if n = 1, 2 and p = 3 if ; m, r ≥ 1 if n = 1, 2 and 1 ≤ m, r ≤ 5 if n = 3. One can easily verify that u f1 (u, v) + vf2 (u, v) = (p + 1) F (u, v) , ∀ (u, v) ∈ R2 . (1.3) Hao, Zhang and Li [6] .