Classical parabolic theory (PDEs)

Question

I am reading the article Asymptotics of solutions to the periodic problem for a Burgers type equation by Pavel I. Naumkin andd Cristian Jesus Rojas-Milla. I'm almost done, but I don't understand this argument on page 8:

It is known from the classical parabolic theory that in the case when the nonlinearity grows no faster than quadratically in the gradient, it is sufficient to prove $L^{\infty}$- apriori estimate to guarantee global in time existence for the solutions

My questions are:

1. What does "the nonlinearity grows no faster than quadratically in the gradient" mean?
2. Why does the above argument guarantee global solutions?

Answer 1

A nonlinearity is said to grow no faster than quadratically if $$ |f(u)| \le A + C |u|^2 $$ I suggest you read the the references the authors give.

Typically, the solution of parabolic equations are easy to show to exist locally in time (fixed point argument) whereas local existence is trickier (solutions might blow up in norm). Think, for the case of ODEs, in $x' = x^2$. Local solution but blow up in finite time.

If you manage to show that the solution does not blow up in finite time (this understood typically in the sense of norms), then there is global existence. Again, take a look at there references (specially Amann's book [1]).

Cheers.