In the book of Volpert on *Traveling wave solutions of Parabolic Systems* (AMS), one reads "*the following assertion is readily proved and we shall not discuss it in detail*". The same result is tacitely assumed in Evans book on partial differential equations when dealing with traveling wave solutions of the bistable equation.

**Proposition:** If a function $w:\mathbb{R} \rightarrow\mathbb{R}$ in $C^2(\mathbb{R})\cap
C^1_b(\mathbb{R})$ satisfies the ODE
\begin{equation}
w''+\sigma w'+f(w)=0\qquad \text{and}\qquad\lim_{t\rightarrow \pm\infty}w(t)=w_\pm\in\mathbb{R}
\end{equation}
then there exist (and are zero) the two limits
\begin{equation}
\lim_{t\rightarrow \pm\infty}w''=\lim_{t\rightarrow \pm\infty}w'=0 .
\end{equation}

PS the hypotheses on $f$ are not explicitely written.

Can someone give me a reference for that kind of results?

Thanks in advance,
Josh.