Blow up of solutions to parabolic PDE

Question

I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice text explaining this. I'd like to see the calculations done.

Thanks.

Answer 1

Certainly the size of the domain can play a role. For the classical case $$ \partial_t u=\partial_x^2 u+u^2, \text{ $x$ in }[-L,L] $$ with Dirichlet BC I recomend Evan's book (Chapter 9, I think). in this case the condition for blow up is that the initial data projected on the first eigenfunction should be larger that the first eigenvalue. As the eigenvalue/eigenfunction depends on the size of the domain this size plays a role.

For another example, let's define $$ \sqrt{-\partial_x^2}u(x)=\int_{-L}^L \frac{u(x)-u(x-y)}{y^2}dy. $$ If we consider a $L-$periodic solution to $$ \partial_t u=-\sqrt{-\partial_x^2} u+u^2, \text{ $x$ in }[-L,L] $$ with positive initial data of fixed mean equal to 1, we have $$ \frac{d}{dt}\|u\|_{L^\infty}\leq (1-C/L)\|u\|_{L^\infty}^2, $$ and we obtain that hte size plays a role again.

I don't know if this clarify something.

PS: I also put the answer in https://mathoverflow.net/questions/141802/blow-up-of-solutions-to-parabolic-pdes