TITLE: Definition of 'blow up' in the context of PDEs
QUESTION [5 upvotes]: What exactly is 'blow up'? Is there a proper well-defined definition for this term? What does it means mathematically? Does it implies 'infinity'?

REPLY [5 votes]: Let me show it with a particular example. Consider the semilinear heat equation
$$
u_t-\Delta u=u^p,\quad x\in\Omega\subset\mathbb{R}^n,\quad t>0,
$$
with initial value $u(x,0)=u_0(x)\ge0$ and Dirichlet boundary conditions $u(x,t)=0$ for $x\in\partial \Omega$, $t\ge0$. Here $\Omega$ is a bounded smooth domain and $p>1$. Then, for appropriate $u_0$, there is a unique solution $u(x,t)$ defined on a maximal time interval $[0,T)$, $T>0$. If $T=\infty$ the solution is said to be global. If $T<\infty$ then
$$
\lim_{t\to T^-}\sup_{x\in\Omega}u(x,t)=\infty,
$$
and the solution is said to blow-up at finite time $T$.
This can be used ion other equations or systems of PDE's.