Longtime Behavior of Wave Equation with Kinetic Boundary Condition

Abstract

In this paper, we consider a damped wave equation with mixed boundary conditions in a bounded domain. On one portion of the boundary, we have kinetic boundary condition: $$\partial _\nu y+m(x)y_{tt}-\Delta _{T} y=0$$ with density function m(x), and on the other portion, we have homogeneous Neumann boundary condition: $$\partial _\nu y=0$$ . Based on the growth of the resolvent operator on the imaginary axis, solutions of the wave equations under consideration are proved to decay logarithmically. The proof of the resolvent estimation relies on the interpolation inequalities for an elliptic equation with Steklov type boundary conditions.