Decay estimates for Beam equations with potentials in dimension three

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{tt}+({\mathrm{\Delta }}^2+V)u=0,u(0,x)=f(x),u_t(0,x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on ${\mathbb{R}}^3$. Assume that zero is a regular point of $H:={\mathrm{\Delta }}^2+V$, we first prove the following optimal time decay estimates of the solution operators \begin{equation*} \big|\cos (t\sqrt{H})P_{ac}(H)\big|{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{3}{2}}\ \ \hbox{and} \ \ \Big|\frac{\sin(t\sqrt{H})}{\sqrt{H}} P{a c}(H)\Big|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{1}{2}}. \end{equation*} Moreover, if zero is a resonance of $H$, then time decay of the solution operators above also are considered. It is noticed that the first kind resonance does not effect the decay rates for the propagator operators $\cos (t\sqrt{H})$ and $\frac{\sin (t\sqrt{H})}{\sqrt{H}}$, but their decay will be dramatically changed for the second and third resonance types.