Decay estimates for beam equations with potentials on the line

TL;DR

This paper analyzes the time decay of solutions to the 1D beam equation with a decaying potential $V$ by modeling the spatial operator as $H=\Delta^2+V$ and projecting onto the absolutely continuous spectrum. Using Stone's formula, Littlewood-Paley decomposition, and detailed low- and high-energy resolvent analysis, it establishes $L^1\to L^{\infty}$ dispersive bounds for the ac part of $\cos(t\sqrt{H+m^2})$ and $\frac{\sin(t\sqrt{H+m^2})}{\sqrt{H+m^2}}$, with sharp behavior: for $m=0$ the decay is $|t|^{-1/2}$, while for $m\neq0$ the low-energy part yields $|t|^{-1/4}$ and the high-energy part yields $|t|^{-1/2}$, and these bounds are uniform with respect to zero being regular or resonant. The results extend to Strichartz estimates, enabling nonlinear beam equation analysis, and the paper carefully treats zero-energy resonances (regular, first-kind, second-kind) via expansions of $(M^{\pm}(\lambda))^{-1}$, under precise decay rates on $V$. Overall, the work shows that the dispersive behavior aligns with the free case across resonance scenarios and provides a robust framework for nonlinear applications.

Abstract

This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( Δ^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = Δ^2 + V$ 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.

Theorems & Definitions (34)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3