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.
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.