Q:

Lipschitz constant for a second order nonlinear differential equation

I'm trying to calculate the Lipschitz constant for a second order nonlinear differential equation:
$$y'' + y' + y^n = 0, \; y(0)=0, \; y'(0)= 0 \text{ and } n>1$$
Should I solve for $y(x)$ and differentiate the solution to find a bound ($L$, Lipschitz constant) on $\vert f'(x) \vert$? Or does it mean that I have to differentiate $f(x)= y'' + y' + y^n$? I would like some tips to proceed as this is my first calculation of this kind. Thanks in advance.

A:

To apply the formalism you use the Lipschitz constant for, you first need to transform the ODE into a first order system,
\begin{align}
y'&=v\\
v'&=-v-y^n.
\end{align}
Next we know that non-linear polynomials are never globally Lipschitz, you only get a local Lipschitz constant on a set like $|y|<R$. Then
\begin{align}
\|F(y_2,v_2)-F(y_1,v_1)\|_{\max}&=\max(|v_2-v_1|,|v_2-v_1+y_2^n-y_1^n|)\\
&\le |v_2-v_1|+nR^{n-1}|y_2-y_1|\\
&\le (1+nR^{n-1})\|(y_2,v_2)-(y_1,v_1)\|_{\max}
\end{align}
This is sufficient to prove uniqueness, but only local existence. No claim on the maximal domain follows from this bound.

