Some decay properties for the damped wave equation on the torus

On some properties of the Lyapunov equation for damped systems

We consider a damped linear vibrational system whose dampers depend linearly on the viscosity parameter v. We show that the trace of the corresponding Lyapunov solution can be represented as a rational function of v whose poles are the eigenvalues of a certain skew symmetric matrix. This makes it possible to derive an asymptotic expansion of the solution in the neighborhood of zero (small dampi...

Achieving Arbitrarily Large Decay in the Damped Wave Equation

is referred to as the decay rate associated with a. If a is to be introduced in order to absorb an initial disturbance then one naturally wishes to strike upon that a with the least possible (most negative) decay rate. The mathematical attraction here lies in the oftnoted fact that, with respect to damping, ‘more is not better.’ More precisely, for constant a, the decay rate is not a decreasing...

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation utt −∆u+ a(1 + |ut|)ut = bu|u|p−2, a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponenti...

Local decay for the damped wave equation in the energy space

Energy decay for the damped wave equation under a pressure condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by...