Long Time Behavior Of Solutions To Nonlinear Diffusion Equations Coupled Via Nonlinear Boundary Sources

In this thesis, we investigate the long time behavior of solutions to multidimensional nonlinear diffusion equations with nonlinear boundary sources. This thesis is divided into three chapters.In chapter 1, we introduce the related works to the problem considered and our main results.Chapter 2 is concerned with one prototype of nonlinear diffusion equations coupled by the nonlinear boundary sources on the exterior domain of the unit ball in RN as follows, where m,n>1, αi, βi≥0,i=1,2, N>2,B1(0) is the unit ball in RN with boundary (?)B1(0), v is the inward normal vector on (?)B1(0), and u0(x),v0(x) are nonnegative, suitably smooth and bounded functions with compact supports. By constructing several kinds of sup-solutions and sub-solutions, we obtain that, under some assumptions, the global existence critical curve coincides with the Fujita critical curve for the system.In chapter 3, we are concerned with non-Newtonian filtration equations coupled by the non-linear boundary sources on the exterior domain of the unit ball in RN as follows, where p,q>2, αi, βi> 0,i=1,2, N>2, B1(0) is the unit ball in RN with boundary (?)B1(0), v is the inward normal vector on (?)B1(0), and u0(x),v0(x) are nonnegative, suitably smooth and bounded functions with compact supports. Similar to the proofs of chapter 2, we get the sufficient conditions of existence of global solutions and the blow-up solutions of the system.