Study On Properties Of Solutions To Some Nonlinear Parabolic Equations(Systems)

As an important class of partial differential equations, nonlinear parabolic equation comes from many disciplines such as physics, chemistry,dynamics of biological groups, and we can find the specific mathematical models in the field of fluid mechanics, theory of phase transitions, image processing etc. Therefore, the studying on the properties of solutions to nonlinear parabolic equation can explain many natural phenomena, which also enriches enormously the theories of partial differential equations.This dissertation mainly studies the properties of solutions to some classes of nonlinear degenerate parabolic equations(systems). Under different boundary value conditions, we discuss the profiles of solutions to parabolic equations(systems) with different source terms, which include the local existence, the global existence, finite time blow-up and asymptotic behavior.It is divided into eight chapters.Chapter 1 summarizes the background of related issues, and briefly introduces the main work of the present thesis.In chapter 2, we investigate the properties of solutions to a degenerate parabolic equation under the homogeneous Dirichlet boundary condition,and the source term is the product of the local form and norm-type. The regularization method is used to show the local existence of nonnegative weak solutions.By comparison principle,we prove the blow-up conditions of nonnegative solutions. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.Chapter 3 deals with the blow-up and global existence of solutions of a degenerate parabolic system with nonlinear norm-type sources. By upper and lower solution methods and comparison principle, we give the criteria for global existence or finite time blow-up to solutions.In chapter 4, applying the similar method and skill in the last chapter,we concern the solutions to a class of nonlinear parabolic system under the homogeneous Dirichlet boundary conditions, which expands a single equation of the chapter 2 into n, and the critical blow-up exponent of the system is gained.In chapter 5, we deal with the blow-up problems of the positive solutions to a degenerate parabolic equation not in divergence form with nonlocal source and nonlocal nonlinear boundary condition. We show how the diffusion coefficient 、the weighted function in the boundary and the nonlinear index affect the properties of solutions. The blow-up and global existence conditions are obtained. For some special cases, we also give the blow-up rate estimates.Chapter 6 extends the results of the last chapter into the case of parabolic system. Similar to the process that we study the single equation,we prove the global existence and nonexistence of nonnegative solutions by constructing some proper super(sub) solutions. In addition, we establish the blow-up rate estimates for some special cases.In chapter 7, we investigate the conditions for finite time blow-up or global existence to a degenerate parabolic system with norm-type sources and weighted nonlocal boundary conditions. We show the weight functions play the substantial roles in determining whether the solutions will blow-up or not. And then we establish the blow-up rate estimates.In the last chapter,we deal with the blow-up problems to a class of degenerate parabolic equations coupled withpL- norm type nonlinear reactions subject to positive value boundary conditions. The blow-up criteria and the global existence of nonnegative solutions are determined,and the results show that the positive value boundary play a key role in determining the blow-up of solutions.