非局部扩散方程行波解定性性质的研究

In recent years,a lot of nonlocal dispersal equations have been derived from the research in many disciplines, such as material science, biology,epidemiology and neural network. Although the nonlocal dispersal represented by the integral operator is closer to the reality, it leads to the many new mathematical difficulties and the essential change of dynamics. For example, the solution semi-flows are not usually compact，and the solutions do not have a priori regularity.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 qualitative properties of traveling wave solutions for nonlocal dispersal equations with nonlocal reaction terms, such as the existence、asymptotical behavior and uniqueness. When the nonlinearity term satisfies some monotone condition, the uniqueness and monotonicity of traveling wave solutions are established by considering the exponentially asymptotical behavior of traveling wave solutions at infinite point. When the nonlinearity does not satisfy the monotone condition, the comparison principle fails for this equation. All the methods avaliable to obtain the existence of the monotone traveling wave solutions can not applied directly. In order to solve this problem, we need to find a new method, which does not depend on the comparison principle.

近年来，在材料科学、生态学、流行病学、神经网络等学科中导出了许多非局部扩散方程，并已得到人们的广泛关注。我们知道，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，由于非局部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。这给数学理论的研究带来了新的困难。 在非局部扩散方程的研究中，行波解是一个重要分支，它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。本项目将研究具有非局部反应项的非局部扩散方程行波解的定性性质，主要包括单调行波解的存在性、渐近行为和唯一性。当非线性项满足一定的单调条件时，通过考察行波解在正负无穷两端的指数渐近行为，建立行波解的唯一性和单调性；当非线性项不满足单调条件时，比较原理不再成立。现有的建立单调行波解存在性的方法都不能直接用来得到其存在性。我们将寻找不依赖比较原理的方法来加以研究。

行波解是非局部扩散方程研究的一个重点。本项目针对两类非局部扩散方程的行波解的定性性质进行了深入的研究。对于具有非局部反应项的非局部扩散方程，我们首先建立了行波解的渐近行为，进而得到了行波解的唯一性；在出生函数非单调的假设下，我们得到了极小波速行波解的存在性，并证明极小波速与渐近传播速度相一致；最后证明了大波速行波解的稳定性；对于具有年龄结构的非局部扩散种群模型，我们证明了大波速行波解的稳定性。