Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 91, pp. 1-33.

Title: Existence and blow up in a system of wave equations with nonstandard nonlinearities
       
Authors: Salim A. Messaoudi (Univ. of Sharjah, United Arab Emirates)
         Oulia Bouhoufani (Univ. Batna-2, Algeria)
         Ilhem Hamchi (Univ. Batna-2, Algeria)
         Mohamed Alahyane (Univ. of Sharjah, United Arab Emirates)

Abstract:
In this article, we consider a coupled system of two nonlinear hyperbolic equations,
where the exponents in the damping and source terms are variables.
First, we prove a theorem of existence and uniqueness of weak solution,
by using the Faedo Galerkin approximations and the Banach fixed oint theorem.
Then, using the energy method, we show that certain solutions with positive initial
energy blow up in finite time.
We also give some numerical applications to illustrate our theoretical results.

Submitted March 16, 2021. Published November 16, 2021.
Math Subject Classifications: 35D30, 35B40, 35B44, 35L70.
Key Words: Hyperbolic system; existence; blow up, variable exponents; nonlinear.