Decay estimates for the two-dimensional Beam equation with potentials
Abstract
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√HPac(H)|L1→ L∞ ≲ 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.
Related
- Optimal time decay of the Boltzmann equation with cutoff soft potentials in the whole space
- Sharp Decay estimates for the discrete Fourth-Order Schrödinger Equation
- Carleman estimates for a nonconservative Timoshenko beam involving Dirichlet-Neumann boundary conditions and application to exact controllability
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Decay estimates for Yang-Mills connections and Yang-Mills-Higgs pairs in higher dimensions
BibTeX
@misc{chen2026decay,
title = {{Decay estimates for the two-dimensional Beam equation with potentials}},
author = {Shuangshuang Chen and Han Cheng and Zijun Wan and Xiaohua Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.16793},
eprint = {2606.16793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16793},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decay estimates for the two-dimensional Beam equation with potentials
AU - Shuangshuang Chen
AU - Han Cheng
AU - Zijun Wan
AU - Xiaohua Yao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2606.16793
UR - https://arxiv.org/abs/2606.16793
AB - This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential V: ∂_t^2 u + (Δ^2 + 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=Δ^2+V, we first prove sharp L^1→ L^∞ estimates for the solution operators: |cos(t√H)P_ac(H)|_L^1→ L^∞ + |sin(t√H)/t√HP_ac(H)|_L^1→ L^∞ ≲ 1/|t|, and obtain an enhanced decay (|t|log|t|)^−1 in logarithmically weighted spaces L^1_ω→ L^∞_−ω with ω(x)=log(2+|x|). For second-kind resonances of H (the bi-Laplacian Δ^2 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 L^1→ 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 L^1→ 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 L^1→ L^∞ estimate |t|^−1.
ER -