Solution of a differential equation having a singularity (not everywhere defined) Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity).

I want to remember the exact definition of a solution with singularity, which I studied in a university long time ago.

And please, an example.
The easiest example is the `finite time blow-up' ODE: 

$$
y^\prime =y^2,\qquad y(0)=1
$$

This equation has the unique solution 

$$
y(t)=\frac{1}{1-t}
$$ which is not defined when $t=1$.  So technically speaking, $y(t)$ isn't a solution to $y^\prime=y^2$ for all time - we have to choose an interval where the solution makes sense.  The rigorous approach would go as follows: 

We say that $y(t)$ is a (classical) solution to the following ODE

$$
y^\prime=f(t,y),\qquad y(0)=y_0,\qquad a&lt;t&lt;b
$$ If 


$y^\prime(t)$ exists for all $a&lt;t&lt;b$ (note this requires that $y(t)$ itself exists for all $a&lt;t&lt;b$)
$y^\prime(t)=f(t,y)$ for all $a&lt;t&lt;b$
$y(0)=y_0$


Any solution which fails to satisfy these conditions at some point could be called a `singular' solution.  The example $y^\prime=y^2$ violates the first condition: $y^\prime$ doesn't exist when $t=1$.  However, if we restrict the problem to read as follows: 

$$
y^\prime=y^2,\qquad y(0)=1,\qquad 0\leq t&lt;1
$$ Then $y(t)=(1-t)^{-1}$ is a classical solution.  Note that I have included the left endpoint assuming the appropriate definition of the derivative at $t=0$.  

There is much more - I recommend picking up a book on ODE theory e.g. Hartman or Arnold.
A solution $x(t)$ of the ODE is called maximal if it is defined on an
  open interval and cannot be extended to any larger open interval.


From "Ordinary Differential Equation". Alexander Grigorian. University of Bielefeld. Lecture Notes, April - July 2008

So the things I search for are maximal solutions.

The question remains: What is the best generalization of this for PDE?