Fast energy decay for 2-D wave equation with localized damping near spatial infinity

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.