On the system of $2$-D elastic waves with critical space dependent damping

We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as time goes to infinity. In the $2$-D case we do not have any suitable Hardy type inequality, so generally one has no idea to establish optimal energy decay. We develope a special type of multiplier method combined with some estimates brought by the $2$-D Newton potential belonging to the usual Laplacian $-\Delta$, not the operator $-a^2\Delta - (b^{2}-a^{2})\nabla {\rm div}$ itself. The property of finite speed propagation is important to get results for this system.