Sharp polynomial decay rates for the damped wave equation on the torus

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...

Nearly a polynomial decay rate for the dissipative wave equation

The study of stabilization of the linear dissipative wave equation in a bounded domain with Dirichlet boundary condition is now an old problem. The exponential decay rate of the energy was established by Bardos, Lebeau and Rauch [ BLR] under a geometrical hypothesis linked with the geodesics. Furthermore such condition called geometric control condition is almost necessary to get a uniform expo...

Polynomial decay rate for the dissipative wave equation

This paper is devoted to study the stabilization of the linear wave equation in a bounded domain damped in a subdomain when the geometrical control condition (see [ BLR]) of the work of C. Bardos, G. Lebeau and J. Rauch is not fulfilled. In such case, they [ BLR] proved that the uniform exponential decay rate of the energy cannot be hoped due to the existence of a trapped ray that never reaches...

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