# Blow up of solutions to parabolic PDE

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.

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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.