Decay estimates for Beam equations with potential 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$u_{tt}+({\Delta }^2+V)u=0,u(0,x)=f\left(x\right),u_t(0,x)=g\left(x\right)$ in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of $H={\Delta }^2+V$, we first prove the following optimal time decay estimates of the solution operators${\Vert \cos \left(t\sqrt{H}\right)P_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{3}{2}}\text{and}{\Vert \frac{\sin \left(t\sqrt{H}\right)}{\sqrt{H}}{P}_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{1}{2}}.$ Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators $\cos \left(t\sqrt{H}\right)$ and $\frac{\sin \left(t\sqrt{H}\right)}{\sqrt{H}}$, but their decay will be significantly changed for the second and third-kind resonances.