If you have a local solution to a parabolic PDE (say we know it exists (weakly anyway) from time 0 to T), then if the solution is bounded in an appropriate way (in which norms?) then we can apparently extend the solution globally. Can someone refer me to these results or explain this, please?
I also heard that the as the solution $u(t)$ converges a $C^\infty$ function as $t \to T$. Why is that? I thought this had something to do with Sobolev embeddings but I can't see in general how this latter statement can be true (maybe it's just for this case).