非单调的时滞非局部扩散方程和系统的行波解

In recent years, a lot of nonlocal dispersal equations and systems have been derived from the research in many disciplines, such as material science, biology and epidemiology. Compared to the classical diffusion, the nonlocal dispersal represented by the integral operator is closer to the reality. Howerer, the nonlocal dispersal leads to the essential change of dynamics of equations. For example, the solution semi-flows are not usually compact，and the solutions do not have a priori regularity. Thus, it brings many new mathematical difficulties. In the study of nonlocal dispersal equations, one important topic is their traveling wave solutions, which can well model the oscillatory phenomenon and the propagation with finite speed of nature. This project will investigate the existence, uniqueness and stability of traveling wave solutions of non-monotone nonlocal dispersal equations and systems with delay. We first consider the existence of non-monotone traveling wave solutions of delayed nonlocal dispersal equations, then establish the precise asymptotic behavior of traveling wave solutions at infinity. On this basis, we prove the uniqueness of traveling wave solutions by sliding method. Furthermore, we apply the improved technique weighted energy method to show the stability of non-monotone traveling waves. For the nonlocal dispersal competitive systems, including monostable and bistable cases, we establish the uniqueness and stability of traveling wave solutions.

近年来,材料科学、生态学和流行病学等学科中导出了许多非局部扩散方程和系统.相比于Laplace算子所刻画的经典扩散,用积分算子所表示的非局部扩散能够更加准确地描述所考虑的实际问题.然而,非局部扩散项的出现导致方程的性质和动力学行为发生了改变,例如方程的解半流不再是紧的以及解的正则性降低等,这给其数学理论的研究带来了新的困难.在非局部扩散方程和系统的研究中,行波解是一个重要分支,这是因为行波解可以很好地描述自然界中大量有限速度传播问题及振荡现象.本项目将研究非单调的时滞非局部扩散方程和系统的行波解的存在性、唯一性和稳定性.我们将首先考虑时滞非局部扩散方程的非单调行波解的存在性,然后获得行波解在实轴两端的精确渐近行为,进而利用滑动平面方法得到行波解的唯一性.与此同时,利用改进的加权能量方法,建立非单调行波解的渐近稳定性.对于非局部扩散竞争系统,包括单稳和双稳情形,建立行波解的唯一性和稳定性.

近年来,材料科学、生态学和流行病学等学科中导出了许多非局部扩散方程和系统.在非局部扩散方程和系统的研究中,行波解是一个重要分支,这是因为行波解可以很好地描述自然界中大量有限速度传播问题及振荡现象.本项目主要研究了非单调的时滞非局部扩散方程和竞争系统的行波解的存在性、唯一性和稳定性.首先证明了非单调时滞非局部扩散方程的非单调行波解的存在性,然后利用Ikehara定理获得行波解在实轴两端的精确渐近行为,进而利用滑动平面方法得到行波解的唯一性.对于双稳非局部扩散竞争系统,建立了行波解的渐近行为和波速的唯一性.对于空间离散的单稳反应扩散方程和竞争系统，利用改进的加权能量法得到了行波解的稳定性。