I am reading an articule(http://link.springer.com/content/pdf/10.1007%2Fs00028-010-0085-8.pdf) so I almost done but I dont understand the next argument is in the 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" the questions are: 1) whats means the nonlinearity grows no faster than quadratically in the gradient ? 2) why the below argument guarantee global solutions?

note: i cant get the references... :(

Thank for your answers !