升程阶跃函数
数学
独特性
数学分析
双稳态
反应扩散系统
初值问题
组合数学
功能(生物学)
常量(计算机编程)
数学物理
物理
量子力学
生物
进化生物学
程序设计语言
计算机科学
出处
期刊:Siam Journal on Mathematical Analysis
[Society for Industrial and Applied Mathematics]
日期:1983-11-01
卷期号:14 (6): 1107-1129
被引量:35
摘要
The pure initial value problem for the bistable reaction-diffusion equation \[ v_t = v_{xx} + f(v) \] is considered. Here $f(v)$ is given by $f(v) = - v + H(v - a)$ where H is the Heaviside step function, and $a \in (0,\frac{1}{2})$. It is demonstrated that this equation exhibits a threshold phenomenon. This is done by considering the curve $s(t)$ defined by $s(t) = \sup \{ x:v(x,t) = a\} $. It is shown that if $v(x,0) < a$ for all x, then $\lim _{t \to \infty } \| {v( \cdot ,t)} \|_\infty = 0$. Moreover, there exists a positive constant $c^ * $ such that if the initial datum is sufficiently smooth and satisfies $v(x,0) > a$ on a sufficiently long interval, then $s(t)$ is defined in $\mathbb{R}^ + $, and $\lim _{t \to \infty } (s(t) - c^ * t)$ exists. Regularity and uniqueness properties of $s(t)$ are also presented.
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