2026/06/30 by Shuangshuang Chen, Han Cheng, Zijun Wan +1
Mathematics · #math.AP
65 pages, Some inprovements
arxiv created 2026/07/30 · arxiv updated 2026/07/31
This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential V: ∂t2 u + (Δ2 + V) u = 0, u(0,x)=f(x), ∂t u(0,x)=g(x). When zero is a regular point or a first-kind resonance of H=Δ2+V, we first prove sharp L1→ L^∞ estimates for the solution operators: ‖cos(t√(H))Pac(H)‖L1→ L^∞ + ‖(sin(t√(H)))/(t√(H))Pac(H)‖L1→ L^∞ \lesssim (1)/(|t|), and obtain an enhanced decay (|t|log|t|)-1 in logarithmically weighted spaces L1ω→ L^∞-ω with ω(x)=log(2+|x|). For second-kind resonances of H (the bi-Laplacian Δ2 belongs to this class), a non-zero trace moment ⟨ |x|2V,ϕ⟩≠0 for some second-kind resonance function ϕ induces severe threshold singularities, worsening the L1→ L^∞ estimate to |t|-1(log|t|)2. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of d-wave resonance leads to the worst L1→ L^∞ decay rate ∼(log|t|)-1. Several improved estimates are also obtained without a d-wave resonance. In particular, in the pure eigenvalue case (i.e., neither d-wave nor p-wave resonance), both propagators recover the optimal unweighted L1→ L^∞ estimate |t|-1.