Abstract We are concerned with asymptotic behaviours of solutions for linear wave equations with frictional damping only on Wentzell boundary, but without any interior damping. Making some elaborate and subtle analysis of an associated auxiliary system, we obtain an ideal estimate of the resolvent of the generator of the system along the imaginary axis. This enables us to prove that the energies of the system decay polynomially. Our energy stability result presents a solution, in the linear case, to the problem proposed by Cavalcanti et al. (2007) [8] , which was put forward as a “hard problem” due to the lack of interior damping.

中文翻译：

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具有阻尼 Wentzell 边界条件但无内部阻尼的波动方程解的渐近行为

摘要 我们关注仅在 Wentzell 边界上具有摩擦阻尼但没有任何内部阻尼的线性波动方程解的渐近行为。对相关的辅助系统进行一些精细和微妙的分析，我们获得了系统发电机沿虚轴的解析度的理想估计。这使我们能够证明系统的能量以多项式衰减。我们的能量稳定性结果在线性情况下为 Cavalcanti 等人提出的问题提供了解决方案。(2007) [8] ，由于缺乏内部阻尼而被提出为“难题”。