双稳型反应对流扩散方程的非平面波前解

This project is mainly concerned with the non-planar traveling fronts of reaction-diffusion equations with nonlinear convection in high-dimensional space by using super- and sub-solutions method and comparative principle. More precisely, firstly, we will study the V-shaped traveling fronts of a reaction-diffusion equations with nonlinear convection in two-dimensional space, we construct a V-shaped traveling front, and show that it is the unique V-shaped traveling front between super- and sub-solutions, moreover, we establish a series of estimates and construct different types of super- and sub-solutions to establish the global asymptotic stability of V-shaped traveling front. Secondly, study the asymptotic stability and the instability of V-shaped traveling fronts in dimension n>2, we show that the V-shaped traveling front is asymptotically stable under the initial perturbations decaying at infinity, and the V-shaped traveling front is also asymptotically stable, even if the initial perturbation dose not decay to zero at infinity but the initial value satisfies some certain assumptions, furthermore, we use a counter-example to show that for the general initial perturbation which dose not decay to zero at infinity, the V-shaped traveling front may be unstable. Thirdly, a monotone traveling wave solution which the contour line has a pyramidal shape in three-dimensional space will be studied, we will establish the existence, uniqueness and the asymptotic stability of the pyramidal traveling front in the three-dimensional whole space. Finally, we continue to study the existence, uniqueness and the asymptotic stability of the conical traveling front with radial symmetry about some variables. In this project, the research contents is novel. The results of this project will provide an important theoretical basis in the study of ecology, physics, chemistry and other areas.

本项目主要利用上、下解方法结合比较原理研究一类带有非线性对流项的反应扩散方程在高维空间中的非平面波前解。具体来说，首先，研究方程在二维空间中的V形波前解，建立介于上解和下解之间的唯一V形波前解，进而通过一系列估计并构造不同类型的上、下解建立V形波前解的全局渐近稳定性。其次，研究V形波前解在更高维空间(n>2维)中的稳定性和不稳定性。当初始扰动在无穷远处衰减到零时V形波前解是全局渐近稳定的；当初始扰动在无穷远处不衰减到零，而初值函数满足某些特定假设时，V形波前解仍然是渐近稳定的；对于一般的初始扰动，V形波有可能不稳定。再次，在三维空间中研究方程水平集具有棱锥形的单调行波解，即建立棱锥波的存在、唯一性和稳定性。最后，继续在三维空间中研究方程关于某些变量为径向对称且单调的圆锥形波前解的存在、唯一性和稳定性。本项目研究内容新颖。本项目的研究结果将为生态学、物理、化学等领域的研究提供重要的理论依据。

近年来，非平面波的传播现象已经被发现广泛地存在于种群生态学、化学反应和燃烧理论等学科中，而这些现象都可以通过反应扩散方程的非平面行波解来描述。在反应扩散方程的研究中，行波解是一个重要分支，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。和平面行波解相比，非平面行波解的性质变得更加复杂，但同时也更加具有实际意义。本项目主要利用上、下解方法结合比较原理研究了一类带有非线性对流项的反应扩散方程在高维空间中的非平面行波解。首先，研究了方程在二维空间中的V形波前解，建立了介于上解和下解之间的唯一V形波前解。其次，通过一系列估计并构造不同类型的上、下解建立了V形波前解的全局渐近稳定性。最后，研究了V形波前解在三维空间中的稳定性和不稳定性。当初始扰动在无穷远处衰减到零时V形波前解是全局渐近稳定的；当初始扰动在无穷远处不衰减到零，而初值函数满足某些特定假设时，V形波前解仍然是渐近稳定的；对于一般的初始扰动，V形波前解有可能不稳定。