Return to Question

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, but I think that $f\in C^0_b(\mathbb{R})$ is sufficient.

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

Thanks in advance, Josh.

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: Let $\sigma \in \mathbb{R}$. 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, but I think that $f\in C^0_b(\mathbb{R})$ is sufficient.

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

Thanks in advance, Josh.