Decay estimates for beam equations with potentials on the line

TL;DR

This paper derives decay estimates for solutions to beam equations with potentials on the line, revealing different decay rates depending on the mass parameter and establishing Strichartz estimates for nonlinear analysis.

Contribution

The paper provides new decay estimates for beam equations with potentials, including cases with nonzero mass, and extends results to low and high energy regimes, with implications for nonlinear problems.

Findings

Decay rate of |t|^{-1/2} for massless case

Decay rate of |t|^{-1/4} and |t|^{-1/2} for massive case in different energy regimes

Established Strichartz estimates for nonlinear beam equations

Abstract

This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( \Delta^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = \Delta^2 + V$ 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: $$ \left\|\cos (t \sqrt{H}) P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} + \left\|\frac{\sin (t \sqrt{H})}{t \sqrt{H}} P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} \lesssim |t|^{-\frac{1}{2}}. $$ But for $m \neq 0$, 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.