Sharp traveling waves for degenerate diffusion equations with delay

本研究计划研究由生态学引起的偏微分方程，一类具有时滞的退化扩散方程，它描述了具有年龄结构的单一物种的种群动态，例如澳大利亚的苍蝇。由同种或恶化的环境条件之间的竞争引起的退化扩散是生态学中种群扩散模型的一个共同特征，然而，与研究充分的线性扩散情况相比，我们对退化扩散方程的波动动力学行为的了解还有许多差距。简并性通常导致解失去其规律性，并增加了锐型行波的可能性，其中人口密度在有限点下降到零，而不是渐近衰减到零。另一方面，大的时滞往往导致解振荡。这两个事实使得解的结构更加复杂，相关研究也更具挑战性。本研究的目的是了解时滞和扩散退化对种群动态模型的影响。主要目标包括：一是关于行波的结构。受扩散时滞和简并性的影响，我们对尖锐行波、振荡行波和振荡尖锐波前的分类特别感兴趣。我们将试图证明这些波的存在/不存在，并阐明种群动力系统具有这些波前的标准和机制。第二个问题是试图研究这些波前的稳定性，特别是尖波和振荡尖波。因为尖锐波的结构与单调/非单调行波完全不同，并且这些波前的渐近稳定性是非常具有挑战性的，从未得到过处理。本课题旨在发展和丰富具有时滞的简并扩散方程理论，为研究尖锐波前稳定性引入新的数学思想和方法，并为培养博士生和博士后提供良好的机会。

This research proposal concerns the study of partial differential equations arising from ecology, a class of degenerate diffusion equations with delay, which describe the population dynamics of the single species with age-structure, such as Australian blowflies. Degenerate diffusion caused by competition between conspecifics or deteriorating environmental conditions is a common feature of population spreading modeling in ecology, however, there are numerous gaps in our knowledge about the wave dynamical behaviors of degenerate diffusion equations compared with the well-studied linear diffusion case. The degeneracy usually causes the solutions to lose their regularities, and raises the possibility of sharp type traveling waves, where the population density decreases to zero at a finite point, rather than decaying to zero asymptotically. On the other hand, the large time-delay often causes the solutions to be oscillating. These two facts make the structure of solutions more complicated and the relevant study more challenging.    The purpose of this research project is to understand the effects of time-delay and degeneracy of diffusion for the population dynamic models. The main objectives include: One is about the structure of traveling waves. Affected by the time-delay and degeneracy of diffusion, we are particularly interested in the classification of sharp traveling waves, oscillatory traveling waves, and oscillatory sharp wavefronts. We will try to prove the existence/non-existence of these waves, and clarify what will be the criteria and mechanism for the population dynamic system to possess these wavefronts. The second issue is to try to study the stability of these wavefronts, in particular, the sharp waves and oscillating sharp waves. Because, the structure of sharp waves is totally different from the monotone/non-monotone traveling waves, and the asymptotic stability of these wavefronts is quite challenging and never treated.    The proposed research program is expected to advance and enrich the theory of degenerate diffusion equations with time-delay, to introduce new mathematical ideas and methods to study stability of sharp wavefronts, and also to provide good opportunity to train Ph.D. students and postdocs by carrying out this project.