# Parabolic PDE local and global existence

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).

Thanks.

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As for your second question, perhaps you mean a convergence as $t\to\infty$ of a global solution. In parabolic case this has to do with dissipation but for instance if you have an external forcing generally it is not true.