两类反应扩散方程的非平面行波解

The traveling wave solutions of reaction-diffusion equations can be used to describe the propagating phenomena in many fields such as physics, chemistry and infectious diseases. Research on planar traveling wave solutions began to receive attention from the 1970s. And from the 1990s, research on nonplanar traveling wave solutions began to receive attention along with the gradual maturity of the planar traveling wave theory. However, the theory, especially the theory of nonplanar traveling wave solutions of nonlocal diffusion equations is far from perfect up to now. Thus, the studies of nonplanar traveling wave solutions are very necessary. First, this project wants to establish the strict mathematical theory for the nonplanar propagating phenomena observed in the Belousov-Zhabotinskii (BZ for short) chemical reaction, and investigate other kinds of nonplanar propagating phenomena. And second, the project will study the nonplanar traveling wave solutions of a nonlocal diffusion equation with the bistable nonlinearity. But the degeneracy of the BZ reaction system and the lack of regularity of the nonlocal equaiton result in difficulty of the study on their nonplanar waves. This object will apply upper-lower solution technique, combining with monotone iteration theory, nonlinear functional analysis, regularity theory and asymptotic speed of spread of parabolic equations, to study the existence, qualitative properties of nonplanar traveling wave solutions. Furthermore, the asymptotic stability of such wave solutions will also be studied.

反应扩散方程的行波解可以描述物理、化学和传染病学等众多领域的传播现象。自上世纪70年代起，平面行波解的研究开始受到关注。上世纪90年代以来，随着平面波理论的逐渐成熟，非平面波的理论研究也开始受到重视。但目前为止，非平面波的理论尤其是非局部方程的非平面波的研究还很不完善，因此对其研究很有必要。首先，对于Belousov-Zhabotinskii(简称BZ)化学反应中观测到的非平面波的传播，本项目拟建立其严格的数学理论结果，并从理论上研究其更多可能的传播现象。其次，本项目拟研究双稳型非局部反应扩散方程的非平面波。然而，BZ反应系统的退化性、非局部扩散方程正则性的缺失导致这两类方程非平面波的研究相对困难。本项目将利用上下解技术、结合单调迭代理论、非线性泛函分析、反应扩散方程的正则性理论、渐近波速等理论，研究上述两类反应扩散方程的非平面行波解的存在性及其定性性质，并探讨非平面波的渐近稳定性。

行波解理论是反应扩散方程理论的重要组成部分。目前，平面行波解理论的已非常成熟，而非平面行波解的理论研究应还有待进一步完善。..本项目首先研究了Belousov-Zhabotinskii(简称BZ)化学反应中观测到的非平面波的传播现象，利用上下解方法、单调迭代理论、比较原理并结合挤压技术建立了双稳型BZ反应扩散系统的棱锥形波前解的存在性和稳定性结果，并进一步讨论了双稳型BZ反应扩散系统的圆锥形波前解的存在性及其定性性质。其次，本项目研究了带双稳型非线性项的非局部反应扩散方程的非平面波前解，利用上下解方法、比较原理、非线性泛函分析理论、渐近波速理论等建立了其V形波前解的存在性和稳定性结果，棱锥波和圆锥波的存在性结果并讨论了其一些定性性质。..本项目主要致力于建立两类反应扩散方程非平面行波解的理论结果，所得到的关于Belousov-Zhabotinskii化学反应扩散方程的结果可能对Belousov-Zhabotin skii化学实验提供一定的参考价值。