具非局部項之反應擴散方程的週期行波解

Abstract

本文討論帶有Fisher 型非局部反應項的反應擴散方程，其非局部項由以某種機率 分佈為積分核的摺積組成。為瞭解積分核對方程式解造成的影響，與古典Fisher-KPP 方程的穩定解、行波 解做比較，我們希望能夠構造出不同於古典方程的解。 最後若考慮非對稱的積分核，利用bifurcation 與singular perturbation 方法， 我們可以構造出具週期性的行波解。 為整個理論的完整，在第二節中本文引用前人的方法並做一些改進，證明具一 般積分核的方程式行波解存在性。

In this article, the reaction-diffusion equation arising from population dynamics with Fisher-type non-local consumptions defined through an interaction integral kernel is concerned. In order to know the impact of the integral kernels on the solutions, we try and expect that there exist some non-typical traveling waves different from waves of the classical Fisher equation. Through the bifurcation and perturbation methods, we can generate periodic traveling waves of these equations for the asymmetric integral kernels. By the way, to make the result complete, the existence of solutions for a general class of integral kernel is shown in section 2 through a little modification of methods in the references.