含分数阶拉普拉斯的行波解的研究

Problems involving the fractional Laplacian is one of the hot topics in the current research in partial differential equations. This project will mainly study the traveling wave solutions to the reaction diffusion equation involving the fractional Laplacian. The traveling wave solution is the simplest solution to the reaction diffusion equation, but the study of the traveling wave solution can help understand other solutions’ many properties. First, we are going to study the propagating speed of the solution when the fractional Laplacian is critical and supercritical, when the reaction term is the combustion model, we are going to prove that the spatial variable will super-linearly depend on the time variable, which can give a reason of the nonexistence of one-dimensional traveling wave solution in this case. Secondly, by using the comparison principle and the method of the s-harmonic extension, we plan to study the stability of the one dimensional traveling wave solution in the bistable case, and obtain the uniform bound and convergence between the solution to the initial value problem of the reaction diffusion equation and the traveling wave solution.Finally, we would like to use the sub-super solution method to study the existence of higher dimensional traveling wave solution, especially the existence of the higher dimensional traveling wave solution for the combustion model will be an important topic in the project . In the mean time, we also want to use the moving plane method and s-harmonic extension to study the symmetry of two dimensional traveling wave solution.

含分数阶拉普拉斯的问题是当下偏微分方程研究领域里的热点之一。本项目的研究问题是含有分数阶拉普拉斯的反应扩散方程的行波解。行波解是反应扩散方程的最简单的解，对行波解的研究能帮助了解其他解的性质。首先，我们将研究在分数阶拉普拉斯为临界和次临界情况下的传播速度问题。当反应项函数为燃烧模型时，我们拟证明其空间变量超线性依赖于时间变量，从而解释其一维行波解不存在。其次，我们将利用比较原理和s-调和延拓的办法研究双平稳模型的一维行波解的稳定性，得到反应扩散方程的初值问题解与行波解的差的一致估计和收敛性。最后，我们将利用上下解方法研究高维行波解的存在性,特别是反应项函数为燃烧模型的高维行波解的存在性问题是本项目研究的重点。同时我们也将用移动平面法和s-调和延拓研究二维行波解的对称性问题。

含分数阶拉普拉斯的反应扩散方程的传播问题在众多物理、化学、生物和地质学研究中有重要作用。.. 本项目利用上下解方法研究了含有分数阶拉普拉斯以及非线性反应项为退化Fisher-KPP模型或者燃烧模型的反应扩散方程的传播速度问题。针对退化Fisher-KPP模型，我们得到了传播速度代数依赖于时间的关系，该结果不同于经典拉普拉斯情况的线性依赖关系或者含分数阶拉普拉斯的Fisher-KPP模型的指数依赖关系。我们得到了传播速度的最优上界估计和两个下界估计。针对含临界或超临界分数阶拉普拉斯的燃烧模型，我们得到了传播速度的最优上界估计和最优下界估计。以上结果以论文形式发表在杂志Nonlinearity上。.. 除了以上结果，本项目也在含有分数阶拉普拉斯的二维双平稳模型的非平面行波解的存在性方面取得一些进展。