A Quasilinear Reaction-diffusion System Coupled Via Nonlocal Sources

In this thesis, we mainly consider a quasilinear reaction-diffusion system coupled via nonlocal sources, subject to homogeneous Dirichlet conditions and nonegative initial data. We prove the local existence of the classical solutions as well as the global existence and non-existence of solutions. In particular, we give a precise analysis on the asymptotic behavior of solutions: blow-up rate, blow-up set, and so on. Two kinds of characteristic algebraic systems are introduced to make clear the interaction of all the nonlinear exponents. An interesting phenomenon is observed that the critical exponent is determined by the six nonlinear exponents from all the three nonlinearities, while the blow-up rate is independent of nonlinear diffusion exponents due to the nonlocal sources contained in the reaction terms.We give the background of the parabolic system in the introduction, and review some basic knowledge of the parabolic equation (system) in Chapter 2. In Chapter 3, we treat the local existence of solutions to the parabolic system. In Chapter 4, we establish the critical exponents of the model and get the blow-up criteria for the solutions. In Chapters 5 and 6, we study the singularity more deeply to get the blow-up rate and blow-up set, respectively. The blow-up set consists of the whole domain.