Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that $\xi(0)=\xi'(0)=0$. Show that if $r$ is sufficiently small and $\phi_t$ is the flow of the differential equation $\dot x=-ax+\xi(x)$, then $$\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x))dt$$ is a $C^1$ function.
I am having trouble proving this.
What I have done so far. Since $\phi_t$ is the flow of the DE then $\dot{\phi_t}=-ax+\xi(x)$. Also $\phi_0(x)=x$ by properties of flow. Since $(0,0)$ is a rest point of the given differential equation and the linearization of the system about $(0,0)$ is $\dot x=-ax$ it follows that the rest point $(0,0)$ is asymptotically stable. Thus, $\lim_{t\to \infty} \phi_t(x)=0$. Then I tried to integrate the given function by parts to see if I can explicitly find it. Here is where I got stuck. In particular I don't see how to use the 'vanishes outside the interval $(-r,r)$. (So I guess $\xi$ has compact support $[-r,r]$??. I have used up all the other hypothesis of the problem. (I think??). Should I simply use the definition for derivative instead? Even so, how do I show that the derivative is continuous?
Question: Can anybody give some pointers as to what I am doing wrong/right?. In particular where does having compact support come into play?. Any hints/suggestions?