Existence and uniform decay for a non‐linear viscoelastic equation with strong damping

Abstract

SUMMARYThis paper is concerned with the non-linear viscoelastic equationWe prove global existence of weak solutions. Furthermore, uniform decay rates of the energy are obtained assuming a strong damping ut acting in the domain and provided the relaxation function decays exponentially.

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

“…Now we state a local existence theorem for solutions of the system (1.1) that can be established by combining the arguments in [1,6,27].…”

“…The following initial boundary value problem: 6) has been investigated by many people. When ρ = 0 and there are no dispersion terms, in the absence of strong damping ( γ i = 0), the system has been investigated by several authors and results concerning existence, decay, and blow-up were obtained [1,2,15,22,25,26,28,29,32,36].…”

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

“…Por tanto, la dinámica de los materiales viscoelásticos son de gran importancia e interés ya que tienen amplias aplicaciones en la física y la ingeniería. Más información al respecto, ver [3,6,21].…”

“…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.…”

Blow Up and Global Existence for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Nonlinear Damping and Source terms

“…For the same problem (1.5), in [13], Song and Zhong showed that there were solutions of (1.5) with positive initial energy that blew up in finite time. For more related works, we refer the reader to [14]- [18].…”