Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

Stabilization of the Kawahara Equation with Localized Damping

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the sm...

Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping

In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation utt −∆u+ f(x, t, u) + ∫ t 0 g(t− τ)∆u(τ) dτ + a(x)ut = 0 in Ω× (0,∞). Here the damping term a(x)ut may be null for some part of the domain Ω. By assuming that the kernel g in the memory term decays exponentially, the damping effect allows us to avoid compactness arguments and and to...

Stabilization of a Boussinesq system with localized damping

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damp...

Wave equations with non-dissipative damping

Smooth attractors for the quintic wave equations with fractional damping

Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors...