| This thesis is concerned with a system of quasilinear parabolic equations based on patial differential equation in mathematical biology,with Rosenzweig-Macarthur type reaction.In this system,the diffusion coefficients depend on time and space.Moreover,the boundary conditions are given in the form which may be Dirichlet,Neumann,or Robin type.This paper consists of four chapters,and its main contents are as follows.In Chapter 1 we introduces the background,significance of the research and the sources of the system.In Chapter 2 we study a system of quasilinear parabolic equations,which is Rosenzweig-Maca-rthur model as well,trying to show the existence and uniqueness of a global solution to the parabolic system,using the method of upper and lower solutions.When using the method,we first define the upper and lower solutions of system.Then we construct two sequences from the iteration process which can be proved the existence and monotone property.And the last of this chapter,we show that the sequences converge to the same solution of the system.In Chapter 3,the aim is to show existence of positive quasisolutions of the corresponding elliptic system.First we construct a pair of upper and lower solutions by the eigenvalue problem as initial,then start the iteration process to achieve two sequences which converge to maximal and minimal solution respectively.In case the maximal solution equals to the minimal one,then they both equal to the unique solution of this elliptic system.In Chapter 4 we study the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal-minimal solutions of the elliptic system.We show the two sequences of upper and lower solutions of the parabolic system converge to the solution of the corresponding elliptic system.By then,we have show n the existence and uniqueness of a global solution to the parabolic system and the asymptotic behavior of the time-dependent solution. |
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