Uniform Blow-up Profiles For A Parabolic System With Nonlocal Sources

The main problem studied in this thesis are properties of positive solutions to a semi-linear parabolic system with non-local nonlinear sources and null Dirichlet boundary conditions. We give the analysis about asymptotic property: blow-up rate, blow-up set and blow-up profiles. We first obtain a sufficient condition for blow-up in finite time as well as a necessary condition for simultaneous blow-up, then the blow-up set is obtained. As the main part, we establish uniform blow-up profiles of solutions in the interior to describe the evolution of the boundary layers.In the introduction, we give the practical background of the parabolic system and in Chap- ter 2 review the history and actuality of the reaction diffusion system. In Chapter 3, we will give a sufficient condition for blow-up in finite time, necessary and sufficient conditions for simultaneous blow-up respectively to obtain the uniformly blow-up rates. At the end, we will deal with the blow-up set of the reaction-diffusion system and the estimate of the boundary layers in Chapter 4.