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 on the line: $${\partial }_t^{2}u+\left(\hat{I}{”}^2+m^2+V(x)\right)u=0,u(0,x)=f(x),{\partial }_{t}u(0,x)=g(x),$$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m\in \mathbb{R}$. Let $H=\hat{I}{”}^2+V$ and $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum of $H$. Then for $m=0$, we establish the following decay estimates of the solution operators: $${\Vert \cos (t\sqrt{H})P_{ac}(H)\Vert }_{L^1\to L^{\infty }}+{\Vert \frac{\sin (t\sqrt{H})}{t\sqrt{H}}{P}_{ac}(H)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{1}{2}}.$$ 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 $\cos (t\sqrt{H+m^2})$ and $\frac{\sin (t\sqrt{H+m^2})}{\sqrt{H+m^2}}$ are bounded by $O(|t{|}^{-\frac{1}{4}})$ in the low-energy part and $O(|t{|}^{-\frac{1}{2}})$ 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.

31 Pages. This is a final version published at JDE