We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for all mild solutions using the language and tools of Hilbert complexes. This framework turns out strong enough to conduct our analysis but also general enough to include a number of interesting examples. Some of these are briefly discussed. By a slight modification of the main arguments, we also obtain corresponding decay results for numerical approximations obtained by compatible discretization strategies.

翻译：我们研究以希尔伯特空间中的抽象发展方程表述的阻尼波传播问题。在包括初始值自然相容性条件在内的一般性假设下，我们利用希尔伯特复形的语言和工具，建立了所有温和解的指数衰减估计。这一框架既足够强大以支持我们的分析，又足够通用以涵盖许多有趣的例子，本文简要讨论了其中一部分。通过主要论证的略微修改，我们还获得了由相容离散化策略得到的数值近似解的相应衰减结果。