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:

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