Energy decay rate of the wave equations on Riemannian manifolds with critical potential
In this article the authors prove two decay results for the energy of solutions of the wave-equation-with-potential on a semi-infinite tube over a Riemannian manifold. More precisely, let \((M,g)\) be an \(m\)-dimensional Riemannian manifold and consider a Cauchy problem on \((0,+\infty )\times M\) as follows: \[ \begin{cases} u_{tt}(t,x)-\Delta u(t,x)+V(x)u_t(t,x)=0 & \text{ for }(t,x)\in (0,+\infty )\times M \\ u(0, x)=u_0(x)\text{ and }u_t(0,x)=u_1(x) & \text{ for }x\in M \end{cases} \] with initial values \(u_0\in H^1(M)\), \(u_1\in L^2(M)\) and potential \(V\in L^\infty (M)\cap C^1(M)\) satisfying the bounds \(V_0\left(1+\rho (x)^2\right)^{-\frac{1}{2}}\leqq V(x)\leqq V_1\left(1+\rho (x)^2\right)^{-\frac{1}{2}}\) where \(V_0\), \(V_1\) are constants and \(\rho\) being the Riemannian distance function on \((M,g)\) from a fixed point \(o\in M\). The \textit{energy} of the solution \(u\) is \[ E(t):=\frac{1}{2}\int\limits_{\{t\}\times M}\left(u_t^2+| \nabla u|^2_g\right)\mathrm dg \] where \(\nabla\) is the Levi-Civita connection of the metric. The authors' main result (see Theorem 1.1 in the article) is an estimate \[ E(t)=O(t^{-1+\delta})\text{ as }t\rightarrow +\infty \] where, if \(V_0\geqq 1\), then \(\delta\in (0,1)\) and if \(0<V_0<1\), then \(\delta\in (1-V_0,1)\). Moreover, under some further technical assumptions on the curvature of \((M,g)\) a refined estimate is proved, too (see Theorem 1.2 in the article).
scientific article; zbMATH DE number 6932253
scientific article