TITLE: Estimating rate of blow up of an ODE
QUESTION [9 upvotes]: Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$.
Several related questions: (1) Can I conclude blow up of a solution for some initial conditions? (2) Via taylor series, or whatever relevant approximations of $f(x)$, can we estimate the rate of blow up? And (3), again by approximation, can we estimate given an initial condition when this blow up occurs?
I have in mind $x'=x^p$ which has solution $x=[(1-p)t-C]^{-1/(p-1)}$. There is finite time blow up of order $O(t^{1/(p-1)})$. Given an initial condition $x(0)=X$, we have blow up occur at $X^{1-p}/(p-1)$.

REPLY [8 votes]: With initial condition $x(0)=X$, the solution to your differential equation satisfies $$ \int_X^{x(t)}\frac{dy}{f(y)} = t, $$ as one can check by differentiating both sides and using the chain rule.  The necessary and sufficient condition for blow-up at a finite time $T$, meaning $x(t)\to\infty$ as $t\to T$ from below, is then that 
$$\int_X^\infty \frac{dy}{f(y)}=T.$$
An explicit asymptotic estimate is available for $T$ as $X$ becomes large if $f$ is "regularly varying with exponent $p>1$". This means that $f$ is almost a power law, in the sense that
$$f(y)=y^p L(y) \quad\mbox{ where }\quad \frac{L(cy)}{L(y)}\to 1$$ 
as $y\to\infty$ for every $c>0$.  For then one can deduce that as $X\to\infty$, the blow-up time
$$
T = \int_X^\infty \frac{dy}{y^p L(y)} = 
\frac{X}{X^pL(X)}\int_1^\infty \frac{L(X)}{L(Xz)}\frac{dz}{z^p}
\sim \frac{X^{1-p}}{L(X)} \frac{1}{p-1}.
$$
To justify the last step one needs to use some basic facts proved in books on regularly varying functions, such as those of Seneta or Bingham, Goldie and Teugels. In particular, (i) the convergence  $L(X)/L(Xz)\to1$ occurs uniformly for $z$ in any compact set, and (ii) there is a uniform bound $L(X)/L(Xz)\le Cz^\epsilon$ independent of $X$, for any $\epsilon>0$.