Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up.

Thanks.

Solution 1:

The meaning is, of course, context-dependent...

In the context of differential equations, that a solution to an equation with a "time" variable blows up usually means that the maximal domain for which it is defined is finite, so that at the endpoint of that interval something `bad' happens: either the solution goes to infinity, or it stops being smooth (in a way that makes the differential equation stop having sense, maybe), or something. This is an important phenomenon, one which causes trouble. A couple of examples:

• Perelman's solution of the Poincaré conjecture—in a very vague sense—consists of a way to `work around' the fact that certain solutions of a (very complicated non-linear) PDE blow up;
• the third `Millenium' Clay problem is (very roughly) the question «do the solutions of the Navier-Stokes equation blow up?».

Consider, as a very simple example, the equation $$\frac{\mathrm dx}{\mathrm dt}=x^2.$$ This equation makes sense and satisfies the conditions for existence and uniqueness of solutions on all of the $(t,x)$-plane, but if you solve it (which is easy to do explicitely, as it has the two variabls separated) you'll see that all of its solutions have a maximal interval which is a half-line (which is bounded on the left or on the right, depending on the initial condition) and that at the finite end of that interval the solutions become unbounded. We thus say that all solutions of our equation blow up in finite time.

There are also equations which have some solutions which blow up and some which live forever. One example is $$\frac{\mathrm dx}{\mathrm dt}=\begin{cases}x^2&\text{if $x\geq0$}\\0&\text{if $x\leq0$}\end{cases}$$ and you'll surely find lots of fun in trying to concoct examples where even more interesting phenomena occur.

Solution 2:

It means that as time goes on, the response-- be it displacement, velocity, heat, field or whatever-- will also go to infinity.

This is physically impossible and is usually due to the poor numerical/approximated methods use to solve the PDE.

Solution 3:

It denotes a singularity or discontinuity - as opposed to decaying or stable solutions.

Solution 4:

In the context I usually hear it used, it means that a quantity is undefined or goes to infinity in the limit under consideration. For example, a function which has a singularity at that point blows up at that point (or "as x goes to" that point). Essentially, "blows up" indicates that something is being divided by zero. But I never considered it a technical term, just a bit of mathematical street lingo ;-) I do not know if this is exactly what is meant when it's used in the context of PDE analysis.

By way of example, I would say the integral

$\int_{r_0}^\infty \frac{1}{r^2}\mathrm{d}r$

blows up when $r_0 = 0$. Or that the function $\frac{1}{r^2}$ blows up at $r = 0$.