Decaying properties of the total and local energies for the wave equations with dissipations

In this talk, the dissipative wave equations in an outside of compact regions with smooth boundary are treated. Assume that the dissipation coefficient depends only on the space variable. First, in this setting, a sufficient condition showing uniformly decaying estimates of the local energy is given. As an application of this condition, a decaying estimate of the local energy near the boundary is obtained if the dissipation coefficient is positive near concave part of the boundary in some sense. Second, the cases that the dissipation coefficient is not small in far field are considered. In this case, an optimal decay of the total energy corresponding to the decay rate for high frequency waves is discussed.