Singularity Analysis Of Parabolic Systems Coupled Via Nonlinear Inner Sources And (or) Boundary Flux Of Exponential Types

The main problems studied in this thesis are the properties of the solutions to reaction-diffusion systems coupled via nonlinear sources or(and) nonlinear boundary flux with exponent type, such as global existence, non-global existence, and blow-up rate, etc. Based on observing the relation between the singularity of solutions and the nonlinear mechanism of the systems considered, the so callled linear charateristic algebraic systems are introduced. The solutions of the algebraic systems, or characteristic parameters, are very convenient to describe the critical exponents as well as the propagations of singularities of solutions for the reaction-diffusion systems. For the case with only inner sources or boundary flux coupling, the blow-up rates of solutions are obtained under reasonable conditions, which just are the logarithm of the powers of the characteristic parameters. In particular, the reaction-diffusion systems with localized nonlinear sources are discussed also with a precise analysis to evolution of the boundary layers in the blow-up solutions. The main results obtained in this thesis are summarized as follows:(1) For homogeneous Dirichlet initial-boundary problem about the system through introducing linear algebraic systemwe obtain the critical exponent I(α,β);Under the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·,t) = O(log(T-t)-α), sup v(·,t) = O(log(T-t)-β).(2) For initial-boundary problem ut = △u, vt = △v coupled via boundary conditions (nonlinear boundary flux), we obtain the critical exponent I(α,β) through introducing the linear algebraic systemUnder the radial symmetric condition, we get the blow-p rates and the pointwise estimates of the solutions, thereinto the blow-p rates estimates of the solutions are sup u(·, t) = O(log(T - t)-α), sup v(·, t) = O(log(T - t)-β).(3) For the systemwith homogeneous Neumann boundary conditions, we obtain the critical exponent I(α,β) and blow-up rates through introducing the same linear algebraic system as in (1); At the one-dimensional case, we obtain the pointwise estimates of the solutions and discuss the relation between initail value and the blow-up set.(4) For the system with Neumann boundary conditions ,we introduce some linearalgebraic systemswhere is a matrix, Using the solutions of the linear algebraic systems, we obtain the critical exponent of the problem.The nonlinear parameters in the models considered in our thesis are any real numbers, while the coupling ones among them should be nonnegative.(5) For parabolic systems with localized nonlinear sources, we study the properties of the solutions, such as global existence, non-global existence, asymptotic behavior at blow-up time, and estimates of boundary layer.