2
$\begingroup$

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, preferably with Galerkin method if appropriate.

Thanks.

(CP from https://math.stackexchange.com/questions/483922/blow-up-of-solutions-to-parabolic-pde as I didn't get any attention)

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.