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

We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the results for the two-dimensional exterior mixed problem case, the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincar\'e-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.