Blow-up Estimates of the Positive Solution of a Parabolic System

This paper establishes the blow-up estimates for the systems ut u 0, t 0 in BR Ž0, T., BR n, with the nonlinear boundary conditions u um1 n1 and um2 n2 on SR Ž0, T.where 0 m1 1 m2and 0 n2 1 n1. We prove that cŽT t. 2 max uŽx, t. CŽT t. 2 and cŽT t. 2 max Žx, t. CŽT t. 2 under some monotonicity assumptions on the initial values, where Žn1 n2 1. , Žm2 m1 1. , and n1m2 Ž1 m1.Ž1 n2..