Blow-up Analysis For A Diffusion System With Nonlinear Coupled Boundary Conditions

In this paper, we consider a nonlinear diffusion system with coupled nonlinear boundary flux. Reaction-diffusion system is a special kind of semi-linear parabolic systems, which comes from the widespread existence of a natural phenomenon of the proliferation. The percolation theory, theory of phase change, image processing, biochemistry and biotechnology groups in the fields of the dynamics are all of this kind. We discuss the effect of nonlinear diffusion terms and nonlinear boundary flux to the blow-up property of solutions in detail.In the first chapter, we have introduced some of the background research. Recalling the development of a nonlinear parabolic equations and theoretical work. Also, give a briefing on the main contents of this article.In the second chapter, first, by comparising with an ODE system, using comparison and sub-supsolution methods, we get the blow-up conditions. Then, we constructe an appropriate solution, which globally exists and is a supsolution of the original system, thus, get the global existence conditions. Finally, by using Scaling, we get the blow-up rate under certain conditions.In the third chapter, by researching the blow-up rates of scalar equations, and with the help of the Green function, to a certain condition, we get a necessary and sufficient condition for non-simultaneous blow-up. Then, we have the conditions, such that simultaneous blow-up occur for arbitrary initial data. Next, we proved that the set of initial data such that non-simultaneous blow-up happens is open in L~∞topology. Using this, we discuss the coexistence of both simultaneous and non-simultaneous blow-up, that is, under certain conditions, there exists initial data such that non-simultaneous blow-up occur, also exists initial data such that simultaneous blow-up occur. Finally, using Scaling, we get some conditions, under which non-simultaneous blow-up occur for arbitrary initial data.