Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 73, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.73 EXISTENCE AND DECAY OF SOLUTIONS TO COUPLED SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH VARIABLE EXPONENTS OULIA BOUHOUFANI, SALIM A. MESSAOUDI, MOSTAFA ZAHRI Communicated by Claudianor O. Alves Abstract. In this article, we consider a coupled system of two hyperbolic equations with variable exponents in the damping and source terms, where the dampings are modilated with time-dependent coefficients α(t),β(t). First, we state and prove an existence result of a global weak solution, using Galerkin’s method with compactness arguments. Then, by a Lemma due to Martinez, we establish the decay rates of the solution energy, under suitable assumptions on the variable exponents m and r and the coefficients α and β. To illustrate our theoretical results, we give some numerical examples. 1. Introduction In this work, we study the initial-boundary-value problem u tt - Δu + α(t)|u t | m(x)-2 u t + |u| p(x)-2 u|v| p(x) =0 in Ω × (0,T ), v tt - Δv + β(t)|v t | r(x)-2 v t + |v| p(x)-2 v|u| p(x) =0 in Ω × (0,T ), u = v =0 on ∂ Ω × (0,T ), u(0) = u 0 , u t (0) = u 1 in Ω, v(0) = v 0 , v t (0) = v 1 in Ω, (1.1) where T> 0 and Ω is a bounded domain of R n (n ∈ N * ) with a smooth boundary ∂ Ω; α, β : [0, ∞) → (0, ∞) are two non-increasing C 1 -functions and m, r and p are given continuous functions on Ω satisfying some conditions to be specified later. The wave equations with variable exponents of the nonlinearity occur in math- ematical models of various physical phenomena such as flows of electro-rheological fluids or fluids with temperature dependent viscosity, nonlinear viscoelasticity, fil- tration processes through a porous media and image processing, thermorheological fluids, or robotics, etc. For more details on the subject, the reader can see [1, 10]. In fact, several works concerning hyperbolic problems with nonlinearities of variable- exponent type have appeared, of which we mention some recent ones. 2020 Mathematics Subject Classification. 35L05, 35B40, 35L70, 93D20. Key words and phrases. Coupled system; hyperbolic equation; existence; energy decay; variable-exponent; time-dependent coefficients. ©2023. This work is licensed under a CC BY 4.0 license. Submitted July 29, 2023. Published October 24, 2023. 1