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What exactly is 'blow up'? Is there a proper well-defined definition for this term? What does it means mathematically? Does it implies 'infinity'?

Mathematicing
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  • Yes, it means that the solution goes to infinity inside a compact set, so it's not everywhere defined. – user40276 Apr 23 '15 at 08:38
  • What happens when the solution 'blows up', is 'ill-defined' and 'incomplete'?Do we leave it as it is? When you mentioned "solutions", are you referring to the characteristic lines or some u(x,y......n) function? – Mathematicing Apr 23 '15 at 08:46
  • I'm referring to the solution of the differential equation. Furthermore, it's not ill-defined. Actually, it's well defined inside an open set. The problem is that it's not globally defined since it goes to infinity in the boundary of this open set. – user40276 Apr 23 '15 at 10:40

1 Answers1

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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.

Julián Aguirre
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  • What is the definition then? I've noticed that it is tremendously difficult to look for well-defined definitions and systematic ways to solving PDEs. It is just impossible to. What might you suggest? – Mathematicing Apr 23 '15 at 10:03
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    The definition would be: "the maximal time of existence of the solution is finite and the $L^\infty$ norm of the solutions goes to $\infty$ as $t$ approaches the maximal time of existence." – Julián Aguirre Apr 23 '15 at 12:35
  • @JuliánAguirre In that example, how do you show that it blows up for p sufficiently large without having an explicit solution to work with. Say you just have the mild form. – iYOA Dec 29 '21 at 22:41