Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

Abstract

This work is concerned with a system of nonlinear viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We will prove that the energy associated to the system is unbounded. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. The key ingredient in the proof is a method used in Vitillaro (Arch Ration Mech Anal 149:155-182, 1999) and developed in Said-Houari (Diff Integr Equ 23(1-2):79-92, 2010) for a system of wave equations, with necessary modification imposed by the nature of our problem.

Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms

“…Thereafter, Said-Houari [12] considered the blow-up result for a larger class of initial data with positive initial energy combining potential well method and differential inequality techniques ( [2]). Pişkin [13] studied a coupled semilinear Klein-Gordon system with nonlinear damping terms,…”

“…In the presence of the viscoelastic term (g = 0), Han and Wang [14] discussed semilinear coupled viscoelastic wave system with nonlinear damping terms, (1)- (4) and improved the result in [12] to a larger class of initial data for which the initial energy can take positive values. Besides, for the work on quasilinear wave equations, we refer the reader to [16][17][18] and the references therein.…”

Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping

“…The proof is almost the same that of Said-Houari [26], so we omit it here. □ To prove our result and for the sake of simplicity, we take a = b = 1 and introduce the following:…”

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

“…And this blow-up result was improved by Said-Houari [22] when the initial data are large enough. In [23], Rammaha and Sakuntasathien studied a more general case of (9) by degenerating damping terms.…”