Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

Abstract

SUMMARYIn this paper the nonlinear viscoelastic wave equation in canonical formwith Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set.

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term

“…Lately, these decay results were extended by Messaoudi and Tatar [16] to a situation where a source term is present. Recently, Messaoudi and Tatar [17] studied problem (1.10) for case of γ = 0, they showed that the solution goes to zero with an exponential or polynomial rate under some restrictions on the relaxation function. For other related works, we refer the readers to [18−23] and references therein.…”

No Existencia Global para un Sistema de Ecuaciones de Onda Viscoelástica no Lineal

“…que ha sido estudiado por muchos autores, y sus resultados relativos a la existencia, comportamiento asintótico y explosión de soluciones han sido establecidos recientemente, ver por ejemplo [3,7,10,11,12,16]. Aquí, entenderemos que −∆u , t 0 g (t − s) ∆u (s) ds, − ∆u y f (u) son los términos de dispersión, disipativo de viscoelasticidad, disipativo de viscosidad y fuente, respectivamente.…”

“…Con f (u) = b |u| p−2 u, Messaoudi y Tatar [11] demostraron la existencia global y el decaimiento exponencial. En ausencia del término disipativo de viscosidad y f (u) = b |u| p−2 u, Messaoudi y Tatar [10,12] demostraron la existencia global y el decaimiento exponencial. Con el término disipativo de viscosidad de la forma u y f (u) = |u| p−2 u, Wu [24] demostró la existencia global y los decaimientos exponencial y polinomial.…”

Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

“…In the presence of a nonlinear source term, the decay result has been extended by [23]. In the case of γ = 0 when a source term competes with the dissipation induced by the viscoelastic term, Messaoudi and Tatar [24] studied the equation…”