论文标题
用分数型耗散的胶质波方程的限制行为
Limiting behavior of quasilinear wave equations with fractional-type dissipation
论文作者
论文摘要
在这项工作中，我们研究了Westervelt类型的一类准线波方程，通常是非本地耗散耗散的。它们是通过复杂介质的非线性声音传播的模型而出现的，具有gurtin- pipkin型的异常扩散。针对对所涉及的记忆内核的最小假设（我们允许弱单数），我们证明了在一般的理论框架中此类波动方程的适当性。特别是，我们的结果涵盖了ABEL分数核以及Mittag-Leffler型内核。分析相对于核心依赖的小参数统一进行，并且可以将其物理解释为声音扩散率或热弛豫时间。然后，我们分析解决方案的行为随着该参数消失，并以这种方式将方程与它们的限制对应物相关联。为了确定有限的问题，我们区分不同类别的内核，并分析和讨论随后的所有案例。
In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin--Pipkin type. Aiming at minimal assumptions on the involved memory kernels -- which we allow to be weakly singular -- we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.
