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 $0<a<1$.

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.


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