Mixed Problems for the Wave Equation III. Exponential Decay of Solutions

Let 0 be a bounded object in I? whose boundary F is sufficiently smooth. Concerning the exponential decay of solutions of the wave equation in the exterior of 0 it seems to us that the cases with the Dirichlet boundary condition and with the Neumann, or the third boundary condition are studied. Besides the case with the Dirichlet boundary condition, we know only a few works, for example, Morawetz [11] on the case with the Neumann condition for a convex object and Tokita [13] on the case with the third boundary condition for 0={x\ \x\<^\} . In this paper we suppose the strict convexity of 0, which is an assumption stronger than that of Morawetz [11], and treat the exponential decay of solutions of problems for a very general boundary condition.