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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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up vote 1 down vote accepted

Often you apply a Banach fixed point argument to prove local existence. Then for global existence a sufficient is to show that the norm of the space you used does not blow up in a finite time, assuming a global smooth solution. This is in practically any book that discusses global existence.

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.

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I am also thinking a similar questions since the summer. The answer may depend on your PDE, if the energy is bounded by some constant (which is independent of your local existence time), then you just simply use u(T/2) as your new initial data and you can show the solution exists on [0,3T/2], and so on to get the global extension.

But for the other case, you may need to show your energy is absolutely continuous.

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