It is evident that the (non-trivial) solutions of the simple nonlinear equation $$y'=y^2 $$ all experience a finite time blowup (i.e. their intervals of existence are all bounded). Moreover, if the equation is changed to an inequality as $$y' \geq y^2 $$ the result still holds, as can be seen by separation of variables. I'm interested in some general results about finite time blowups in nonlinear differential inequalities. In particular I'd like a theorem that covers cases like $$|y''| \geq K|y|^{1+\epsilon} $$ with $\epsilon>0$.
Thank you!
For second order equations the situation is not quite the same. If e.g. you look at $$ y'' = -y^3$$ then you may multiply by $y'$ and integrate to get $$ 2 (y')^2 + y^4 = {\rm const}$$ so the solution stays bounded. But must admit that I don't know of a general theory giving conditions for blow-up in the second order case.
You may on the other hand compare with the 'Jacobi-equation' for geodesic flows. It boils down to looking at solutions to $$ y'' + K(t) y = 0 $$ If $K(t)<0$ (negative curvature) there is at most one minimum and if not identically zero the solution goes to infinity at least in one direction of time. My guess is that if you look at an ode like $$ y'' + A(y) y = 0$$ with $A(y)\leq -|y|^\epsilon$ then (if not identically zero) the solution will blow up in finite time at least in one time direction.