In this paper, we consider the Cauchy problem for wave equations with localized damping in Formula: see text. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that Formula: see text as Formula: see text as compared to Nakao’s results in the entire two-dimensional space M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z. 238 (2001) 781–797.. Unlike the results for the two-dimensional exterior mixed problem case (R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, J. Differ. Equ. 188 (2003) 390–405.), the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincaré-type inequalities in all spaces and the finite propagation property of the solution to construct an estimate formula. In the two-dimensional case, when comparing the problem in the whole space with that in the exterior domain, we find that there is a significant difference in the sense that the former requires a logarithmic correction to the energy decay rate.
Ryo Ikehata (Mon,) studied this question.