Decay estimates for beam equations with potentials on the line

Abstract

This paper is devoted to the time decay estimates for the following beam equation with a potential:$\{\begin{array}{cc}{\partial }_t^{2}u+({\Delta }^2+m^2+V(x\left)\right)u=0,& (t,x)\in R\times R,\\ u(0,x)=f\left(x\right),{\partial }_{t}u(0,x)=g\left(x\right),& \end{array}$ where V is a real-valued decaying potential on $R$, and $m\in R$.

Let $H={\Delta }^2+V$ and $P_{ac}\left(H\right)$ denote the projection onto the absolutely continuous spectrum of H. Then for $m=0$, we establish the following decay estimates of the solution operators:${\Vert \cos \left(t\sqrt{H}\right)P_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}+{\Vert \frac{\sin \left(t\sqrt{H}\right)}{t\sqrt{H}}{P}_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{1}{2}}.$ But for $m\ne 0$, the solutions have different time decay estimates from the case where $m=0$. Specifically, the $L^1$-$L^{\infty }$ estimates of $\cos \left(t\sqrt{H+m^2}\right)$ and $\frac{\sin \left(t\sqrt{H+m^2}\right)}{\sqrt{H+m^2}}$ are bounded by $O\left(\right|t{|}^{-\frac{1}{4}})$ in the low-energy part and $O\left(\right|t{|}^{-\frac{1}{2}})$ in the high-energy part.

It is noteworthy that all these results remain consistent with the free cases (i.e., $V=0$) whatever zero is a regular point or a resonance of H. As consequences, we establish the corresponding Strichartz estimates, which are fundamental to study nonlinear problems of beam equations.