Decay property for the linear wave equations in two-dimensional exterior domains. (Q417036)

The paper deals with the initial boundary value problem for the linear wave equation \(u_{tt}-\Delta u=0\), dissipative wave equation \(u_{tt}-\Delta u+u_t=0\) and semilinear equation \(u_{tt}-\Delta u+u_t=| u| ^{p-1}u(1+| x| \log (B| x| ))^{-1}\) in an exterior domain \(\Omega \subset \mathbb {R}^2\) with Dirichlet boundary condition. If the initial data \(u_0,u_1\in H^1(\Omega )\times L^2(\Omega )\) and \(u_1\) (for the wave equation) or \(u_0+u_1\) (for the other equations) belongs to the Hardy space, \(L^2\)-boundedness and then local energy decay of rate \(t^{-1}\) are proved. In the case of semilinear equation small initial data are assumed.