Upper estimates for blow-up solutions of a quasi-linear parabolic equation.

In this paper, we consider a quasi-linear parabolic equation u t = u p (x xx + u) . It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p > 2 . We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2 < p < 3 and in the case of p = 3 , separately. In addition, we assert that if 2 ≤ p ≤ 3 then lim t ↗ T (T - t) 1 p + ε max u (x , t) = 0 z for any ε > 0 under some assumptions. [ABSTRACT FROM AUTHOR]