# bistable equations, Evans PDE, The point which hit the line depends continuously on parameter

In Chapter 4 of Evans PDE, section 4.2.c traveling waves for a bistable equation. It considers the following autonomous first-order system $$\left\{\begin{array}{l}{v^{\prime}=w} \\ {w^{\prime}=-\sigma w-f(v)}\end{array}\right.$$ with $$\lim\limits_{s\rightarrow\infty}(v,w)=(1,0),\ \lim\limits_{s\rightarrow-\infty}=(0,0).$$ Here, $$f$$ satisfies $$\left\{\begin{array}{l}{\text { (a) } f(0)=f(a)=f(1)=0} \\ {\text { (b) } f<0 \text { on }(0, a), f>0 \text { on }(a, 1)} \\ {\text { (c) } f^{\prime}(0)<0, f^{\prime}(1)<0} \\ {\text { (d) } \int_{0}^{1} f(z) d z>0}\end{array}\right.$$ with $$0.

Let $$W^u$$ be the unstable curve leaving $$(0,0)$$ and $$W^s$$ be the stable curve approaching $$(1,0)$$. Let $$L$$ denote the vertical line through the point $$(a+\varepsilon,0$$) with $$\varepsilon>0$$ small. Denote $$W^u\cap L=w_0(\sigma)$$ and $$W^s\cap L=w_1(\sigma)$$.

It says that $$w_0(\sigma)$$ depend smoothly on $$\sigma$$. I don't see how to verify it.