Is there a linear PDE which has blow up solution?

Question

I'm looking for a initial-boundary problem on a compact spatial domain of a PDE such that a unique blow up solution exists. I don't know whether it is possible or not. Can anyone help?

Answer 1

I think the answer really depends on what you understand by blow-up (in other words, which topology you are considering). For instance, if you consider the linear Schrödinger equation $$ i\partial_t u+\Delta u=0,\qquad u\big\vert_{t=0}=u_0, \qquad (t,x)\in\mathbb{R}\times\mathbb{R}^n, $$ you may consider the initial data $$ u_0(x):=\dfrac{e^{-i\vert x\vert^2}}{(1+\vert x\vert^2)^m},\quad \hbox{where }\ \tfrac{n}{4}\leq m\leq \tfrac{n}{2}. $$ By using the explicit form of the solution you can easily check that the following blow-up holds: $$ \vert u(t=1,x=0)\vert=C\int_{\mathbb{R}^n} \dfrac{dy}{(1+\vert y\vert ^2)^m}=+\infty,\qquad C\in\mathbb{R}_+. $$ However, this initial data belongs to $u_0\in C^\infty(\mathbb{R})\cap L^2(\mathbb{R})\cap L^\infty(\mathbb{R})$. Thus, you got a blow-up in (for instance) $L^\infty(\mathbb{R})$. Although, the solution is global in $L^2$.