2021/11/18 by Zhong‐Jie Han, Han, Zhong-Jie, Zhuangyi Liu +3
Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #Lattice Boltzmann Simulation Studies #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2111.09500
openalex publication_date 2021/11/18 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
This paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin--Voigt damping utt(x,t)-[ux(x,t)+b(x)ux,t(x,t)]x=0, x∈(-1,1), tgt;0, where b(x)=0 on x∈ (-1,0], and b(x)=xα>0 on x∈ (0,1) for α∈(0,1). It is known that the optimal decay rate of solution is t-2 in the limit case α=0, and exponential decay rate for α≥ 1. When α∈ (0,1), the damping coefficient b(x) is continuous, but its derivative has a singularity at the interface x=0. In this case, the best known decay rate is t-(3-α)/(2(1-α)). Although this rate is consistent with the exponential one at α=1, it failed to match the optimal one at α=0. In this paper, we obtain a sharper polynomial decay rate t-(2-α)/(1-α). More significantly, it is consistent with the optimal polynomial decay rate at α=0 and the exponential decay rate at α= 1.This is a big step toward the goal of obtaining eventually the optimal decay rate.