Local energy decay for 2-D wave equations with variable coefficients

This paper addresses the two-dimensional initial value problem in ${\bf R}^{2}$ for the wave equation with varying spatial coefficients in the main part. Assuming compactness in the support of the initial value, we report that the corresponding local energy decays to an order of magnitude of, for example, $O(t^{-1}\sqrt{\log t})$ after sufficiently large time. For the two-dimensional whole space case, it is crucial to establish the optimal $L^2$-estimate for the solution itself, skillfully avoiding the difficulty of not being able to use useful inequalities such as Hardy-type inequalities in higher dimensional case. We also consider cases where the variable coefficients are slightly generalized. These proofs are developed using the multiplier method.