Electron. J. Differential Equations, Vol. 2023 (2023), No. 73, pp. 1-20.

Existence and decay of solutions to coupled systems of nonlinear wave equations with variable exponents

Oulia Bouhoufani, Salim A. Messaoudi, Mostafa Zahri

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 \(\alpha(t), \beta(t)\). 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 \( \alpha\) and \(\beta\). To illustrate our theoretical results, we give some numerical examples.

Submitted July 29, 2023. Published October 24, 2023.
Math Subject Classifications: 35L05, 35B40, 35L70, 93D20.
Key Words: Coupled system; hyperbolic equation; existence; energy decay; variable-exponent; time-dependent coefficients.
DOI: 10.58997/ejde.2023.73

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Oulia Bouhoufani
Department of Mathematics
University Batna-2
05000 Batna, Algeria
email: o.bouhoufani@univ-batna2.dz
Salim A. Messaoudi
Department of Mathematics
University of Sharjah, P. O. Box 27272
Sharjah, United Arab Emirates
email: smessaoudi@sharjah.ac.ae
Mostafa Zahri
Mathematics Department
University of Sharjah
United Arab Emirates
email: mzahri@sharjah.ac.ae

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