论文标题
随时间延迟的退化扩散方程的传播速度
Propagation speed of degenerate diffusion equations with time delay
论文作者
论文摘要
我们关注一类退化扩散方程，随时间延迟描述人口动态与年龄结构。在我们最近的研究[{\ em非线性}，33（2020），4013--4029]中，我们确定了时间延迟的退化扩散方程的关键行驶波的存在和唯一性，并获得了关键波速度降低时间延迟的减少机制。在本文中，我们现在能够显示出渐近波的渐近传播速度及其与尖锐波的关键波速度$ c^*（m，r）$的巧合，并证明了溶液在关键波速度$ c^*（m，r）上的初始扰动或紧凑型溶液的紧凑型支撑的边界，用于时间耗尽的时间降低。值得注意的是，与传播速度相关的现有研究不同，时间延迟和退化扩散导致分析扩散速度的某些必不可少的困难，因为时间延迟使行驶波的关键速度放缓，而退化扩散会导致解决方案的规律性丧失。通过相变技术与单调方法相结合，我们可以确定渐近扩散速度。此外，我们提出了一种基于崭新的尖锐差异方案，以处理较大边缘附近的退化扩散$（u^m）_ {xx} $的大变化，并进行了一些数值模拟，这些模拟完美地证实了我们的理论结果。
We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [{\em Nonlinearity}, 33 (2020), 4013--4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed $c^*(m,r)$ of sharp wave, and prove that the initial perturbation or the boundary of the compact support of the solution propagates at the critical wave speed $c^*(m,r)$ for the time-delayed degenerate diffusion equations. Remarkably, different from the existing studies related to spreading speeds, the time delay and the degenerate diffusion lead to some essential difficulties in the analysis of the spreading speed, because the time-delay makes the critical speed of traveling waves slow down, and the degenerate diffusion causes the loss of regularity for the solutions. By a phase transform technique combined with the monotone method, we can determine the asymptotic spreading speed. Furthermore, we propose a brand-new sharp-profile-based difference scheme to handle large variation of degenerate diffusion $(u^m)_{xx}$ near the sharp edge and carry out some numerical simulations which perfectly confirm our theoretical results.
