Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains (Q2747414)

The author of this interesting paper investigates the initial-boundary value problem for the linear strong dissipative wave equation \(u_{tt}(t,x)-\triangle u(t,x)-\triangle u_t(t,x)\), \((t,x)\in (0,\infty)\times \Omega \), \(u(0,x)=u_0(x)\), \(u_t(0,x)=u_1(x)\), \(x\in \Omega \), \(u|_{\partial \Omega }=0\), \(t\in (0,\infty)\), where \(\Omega \subset \mathbb{R}^N\) \((N\geq 2)\) is an exterior domain with compact smooth boundary \(\partial \Omega \) (\(0\notin \bar\Omega \)). Some interesting uniform energy decay estimates of solutions are shown. It is obtained the decay rate such as \((1+t)E(t)\leq K_1+(3/2)K_2\), \(\int\limits_0^tE(s)ds\leq K_3+(1/2)K_2\), \(\|u(t,\cdot)\|^2\leq 2K_2\) (\(K_1,K_2,K_3=\)const and \(E(t)\) is the total energy \(E(t)=(1/2)\|u_t(t,\cdot)\|^2+(1/2)\|\nabla u(t,\cdot)\|^2\)) under some kinds of weighted initial data. The energy decay estimate is achieved by a method based on the combination of the argument due to Ikehata-Matsuyama with the Hardy inequality, which is an improvement of the Morawetz method.