A Free Boundary Problem Arising from a Bistable Reaction-Diffusion Equation
A Free Boundary Problem Arising from a Bistable Reaction-Diffusion Equation
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双稳态反应扩散方程产生的自由边界问题
DOI：
10.1137/0514086
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发表时间：
1983
影响因子：
2
通讯作者：
D. Terman
中科院分区：
文献类型：
--
作者：
D. Terman
考虑一类非线性反应扩散方程\[ v_t = v_{xx} + f（v）\]的纯初值问题.这里$f（v）$由$f（v）= - v + H（v - a）$给出，其中H是Heaviside阶跃函数，$a \in（0，\frac {1}{2}）$。结果表明，该方程具有阈值现象。这是通过考虑由$s（t）= \sup \{ x：v（x，t）= a\} $定义的曲线$s（t）$来完成的。证明了如果v（x，0）a$在一个足够长的区间上，则s（t）$定义在$\mathbb{R}^ + $中，并且存在$\lim _{t \to \infty }（s（t）- c^ * t）$。给出了$s（t）$的正则性和唯一性.
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 $v(x,0) a$ 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.