General Stability in Viscoelasticity

In this chapter, we consider a problem which describes the motion of a viscoelastic body and investigate the effect of the dissipation induced by the viscoelastic (integral) term on the solution. Precisely, we show that, under reasonable conditions on the relaxation function, the system stabilizes to a stationary state. We also obtain a general decay estimate from which the usual exponential and polynomial decay rates are only special cases

Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback

In this paper, we consider the wave equation with a weak internal constant delay term: $$u''(x, t)-\Delta_{x}u(x, t)+ \mu_1(t) \ u'(x, t) + \mu_2(t) \ u'(x, t-\tau) = 0$$ in a bounded domain. Under appropriate conditions on $\mu_1$ and $\mu_2$, we prove global existence of solutions by the Faedo-Galerkin method and establish a decay rate estimate for the energy using the multiplier method

Blowup of solutions of a nonlinear wave equation

We establish a blowup result to an initial boundary value problem for the nonlinear wave equation utt−M(‖B1/2u‖ 2) Bu+kut=|u| p−2, x∈Ω, t>0

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

Abstract In this paper the nonlinear viscoelastic wave equation in canonical form |ut|ρutt − ∆u − ∆utt + � t 0 g(t − τ)∆u(τ )dτ = b|u|p−2u with 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 uniformly decay of solutions provided that the initial data are in some stable set. Keywords and phrases: Global existence, Exponential decay, Nonlinear source, Relaxation function, Polynomial decay, Viscoelasticity

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

Abstract In this paper the nonlinear viscoelastic wave equation in canonical form |ut|ρutt − ∆u − ∆utt + � t 0 g(t − τ)∆u(τ )dτ = b|u|p−2u with 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 uniformly decay of solutions provided that the initial data are in some stable set. Keywords and phrases: Global existence, Exponential decay, Nonlinear source, Relaxation function, Polynomial decay, Viscoelasticity