Classical Parabolic theory (PDE)

Question

I am reading an article and I almost done but I don't understand an 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 $\mathbf L^\infty$ a priori estimate to guarantee global in time existence for the solutions

The questions are:

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

Note: I can't obtain the sources the article cites to compare.

Thanks for your answers !