Properties Of A Parabolic System With Memory Boundary Condition

The study of the blow-up properties for nonlinear parabolic equations has been a important aspect in the field of nonlinear partial differential equations. In this thesis,we investigate the blow-up for two parabolic system coupled with a nonlinear memory boundary flux. We deal with the critiria of blowup, the blow up set and blow up rate of these two system.In Chapter 2,we consider a heat equations with power function type nonlinear memory boundary condition:The main aim of this paper is to study conditions on the global existence or blow-up in finite time of solutions,and the localization of blow up set. Firstly, we establish the comparison principle of this system. Then constructing upper solutions to the equations, we obtain the complete classification for finite time blow-up and global existence. Finally,we also prove that the blow-up will not occur in the interior of the domain.In Chapter 3,we discuss a heat equations with exponent function type memory boundary condition:We first use subtle inequality trick and the properties of Green function to prove that the solutions of equations is blow up in finite time.Secondly, we discuss the blow up set. Finally,we establish the upper bound on the blow-up rate.