Blow-up And Globl Existence OfSoltuion To Non-Networinan Poytropci Filtration System With Multi-coupled Nolinearites

The non-Newtonian polytropic filtration system originates from a variety of diffusion phenomena appeared widely in nature. These equations are proposed in percolation theory, phase transition theory, zoology and other fields of biological groups. In many cases, they are nonlinear and possess degeneration or singularity, therefore, they do not have classical solutions. Properties of solutions can only be discussed in sense of weakness and obtained by using appropriate comparison principle. Research for this kind of equation has formed an important branch of the theory of partial differential equations and has developed intensively.This thesis contains three chapters which deals with the blow-up and global existence of solutions to a non-Newtonian polytropic filtration system coupled with nonlinear source and nonlinear boundary flux.In Chapter One, we give actual background, development history and current situation of the problem that we consider.In Chapter Two, we consider a class of doubly degenerate parabolic system coupled with nonlinear source and nonlinear boundary flux in the one-dimension: the systems and then prove a local existence result by a regularization. Then by a weak comparison principle, necessary and sufficient conditions on the global existence of all positive solutions are obtained.In Chapter Three, we discuss parabolic system coupled with nonlinear source and nonlinear boundary flux in the multi-dimension: Constants satisfy mi , ni > 0,α,β,pi ,qi≥0, i =1,2.Ω∈R N( N≥1) is a bounded domain with smooth boundary denotes the outer unit normal on the boundary , are positive and satisfy the compatibility conditions.First of all, we give the definition of the systems. From the start of weak solutions and a weak comparison principle and by constructing various upper and lower solutions, we obtain sufficient conditions of the global existence of all positive solutions.