If I have a system of $\frac{dy}{dt}=v$ and $\frac{dv}{dt}=-y^3-v$ and I want to show that these solutions satisfy $\frac{d}{dt}(\frac{v^2}{2}+\frac{y^4}{4})\leq 0$ and then use this to establish global existence for $t\geq 0$.
I understand so far that $F(t,y,v) = (v, y^3-v)$ and I solve $F(t,y) = \frac{dy}{dt}=v$ (integrating with respect to t) and get $y(t) = vt + c_1$ and $F(t,v)=\frac{dv}{dt}=-y^3-v$ and $v(t) = -y^3t - vt + c_2$ (integrating with respect to t again).
I believe I am supposed to use the Lipschitz hypothesis and Lipschtiz boundedness to rpove the global existence:
$||F(t,y)-F(t,v)|| \leq L_K||y-v||$
Using this, I get
$||v - (-y^3 -v)|| = ||2v + y^3|| \leq L_K||y-v||$. I am not sure what to do after this step, however, or if I am approaching this problem correctly.
You are to consider how the values of $$E(y,v)=\frac14(2v^2+y^4)$$ change along a solution of $(\dot y,\dot v)=F(y,v)$. The time derivative gives $$ \frac{d}{dt}E(y,v)=v\dot v+y^3\dot y = E'F = v(-v-y^3)+y^3v=-v^2\le 0. $$