2024/12/12 by Shuangshuang Chen, Chen, Shuangshuang, Zijun Wan +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2412.09061
openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: ∂t2 u + ( Δ2 + m2 + V(x) ) u = 0, u(0, x) = f(x), ∂t u(0, x) = g(x), where V is a real-valued decaying potential on ℝ, and m ∈ ℝ. Let H = Δ2 + V and Pac(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: ‖cos (t √(H)) Pac(H)‖L1 → L∞ + ‖(sin (t √(H)))/(t √(H)) Pac(H)‖L1 → L∞ \lesssim |t|-(1)/(2). But for m ≠ 0, the solutions have different time decay estimates from the case where m=0. Specifically, the L1-L^∞ estimates of cos (t √(H + m2)) and (sin (t √(H + m2)))/(√(H + m2)) are bounded by O(|t|-(1)/(4)) in the low-energy part and O(|t|-(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.