Kolmogorov-Fisher 型態的反應擴散方程

Abstract

常係數反應擴散方程的波型解長久以來已被廣泛地研究。Kolmogorov-Fisher型態的方程式有兩種基本型式，其中一種是非線性項有唯一零解的，另一種是非線性項有較高階零解的。這篇論文是在討論方程式u_t=u_{xx}+f(u), x is in (-infty, infty) 的解，當f(u)={e^{-1/u}}*(1-u), f(1)=0, f'(1)< 0 所採用的方法是利用單調性取u,t作為自變數，p(u,t)=u_x(x,t)作為應變數，並對p方程運用下解及上解之概念。

Wavefront solutions of scalar reaction-diffusion equations have been intensively studied for many years. There are two basic cases for the Kolmogorov-Fisher type equations, typified by a nonlinear term with simple zero root and a nonlinear term with higher order zero root. The paper is concerned with solutions u(x,t) of the equation u_t=u_{xx}+f(u), x is in (-infty, infty) in the case f(u)={e^{-1/u}}*(1-u), f(1)=0, f'(1)< 0 The approach is to use the monotonicity to take u and t as independent variables and p(u,t)=u_x(x,t) as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.