Advanced Mathematical Models & Applications Vol.3, No.2, 2018, pp.168-176 EXISTENCE OF GLOBAL SOLUTIONS OF THE MIXED PROBLEM FOR A SYSTEM OF NONLINEAR WAVE EQUATIONS WITH Q- LAPLACIAN OPERATORS Asif Pashayev 1,2 1 Azerbaijan University, Baku, Azerbaijan 2 Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan Abstract. In the paper the mixed problem for the system of nonlinear q-Laplacian wave equations is studied. The theorems are proved on the existence of the global solutions of the considered problem. Keywords: Wave equations, asymptotic behavior, global solutions, initial-boundary value problem, Laplacian operator, Holder’s inequality, Young inequality. AMS Subject Classification: 35L57. Corresponding author: Asif Pashayev, Azerbaijan University, Jeyhun Hajibeyli st.,71, AZ1007, Baku, Azer- baijan, e-mail: asif.pashayev@au.edu.az Received: 03 April 2018; Revised: 10 June 2018 ; Accepted: 18 July 2018 ; Published: 31 August 2018 1 Introduction The study of the existence and asymptotic behavior of global solutions of initial-boundary value problems for the wave equation with nonlinear operators such as u tt − Δ q u +(−Δ) α u t − h (u)= ϕ (t,x) where Δ q u = ∑ ∞ i=1 ∂ ∂x i (   ∂u ∂x i    q ∂u ∂x i ) ,0 <α< 1|h(u)|≈|u| p is investigated in (Gao & Ma, 1999), (Hongjun & Hui, 2007). For this problem, Gao & Ma (1999) (see also (Hongjun & Hui, 2007)) obtained the global existence of the solution when q>p with small initial data when q ≤ p. When q = 2, with the linear damping term α = 0 , Levine (1974) and (Levine, 1974) proved that solution blows up in the finite time with negative initial energy. When q = 2, and the damping term is given by |u t | r u t , r ≥ 0, many authors studied the existence and uniqueness of the global solution and the blowup of the solution, (see (Levine et al., 1997), (Georgiev & Todorova, 1994)). In this paper we consider the nonlinear initial-boundary value problem { u 1tt − Δ q u 1 +(−Δ) α u 1t − f 1 (u 1 ,u 2 )= g 1 (t,x) , u 2tt − Δ q u 2 +(−Δ) α u 2t − f 2 (u 1 ,u 2 )= g 2 (t,x) , (1) u j (t,x)=0,t> 0,x ∈ ∂ Ω ,j =1, 2, (2) u j (0,x)= ϕ j (x) ,u jt (0,x)= ψ j (x) ,x ∈ Ω ,j =1, 2. (3) Here Ω is a bounded domain in R n ,n ≥ 1 with smooth boundary ∂ Ω, t> 0; x ∈ Ω;0 < α ≤ 1,f 1 (u 1 ,u 2 )= a 1 |u 1 | ρ−1 |u 2 | ρ+1 u 1 ; f 2 (u 1 ,u 2 )= a 2 |u 1 | ρ+1 |u 2 | ρ−1 u 2 ;g 1 ,g 1 : [0,T ] × Ω → 168