| The development of the modern science and technology relies on to a great extent the achievements of physics, chemistry, the biochemistry, the engineering science, mathematical economics and so on , in the past several dozens years, along with each domain research thorough development, the science and technology progress already developed unceasingly inseparable, the purification of these discipline also was the factors which make the progress the important guarantee, the precising of disciplines often realized through the establishing mathematical model, but the massive models may sum up as the so-called reaction-diffusion equation .In recent years, many authers have undertaken the research on semilinear elliptic operators on manifolds including the well-known Yamaber problem. In contrast, not much has been done for their semilinear parabolic counterpart. It looks more imperative to fill this gap, when we take into account the tremendoous literature about the heat kernal of a complete Riemannian manifold, The study of Ricci flows also leads to semi-linear parabolic problems. We shall need some new techniques to study when blow-up of solutions occur, and when global postive solutions exist and so on.In the introduction,we give the background to the study of nonlinear reaction-diffusion system on Riemannian manifolds in Chapter 1. and in Chaper 2 we study the global existence and nonexistence of solutions the following homogeneous and inho-mogeneous semilinear parabolic systems:(3.1)(3.2)(3.3)(3.4) where x∈Mn,t > 0 and p,q are positive .At t = 0,nonnegative and continuous initial values (u0(x),v0(x)) are prescribed. As we all known,in the Euclidean case ,(3.I)(3.2) provide a simple example of a reaction-diffusion system, and they can be used as a model to describe heat propagation in a two-component combustible mixture. This article study to the above two kinds of reaction-diffusion systems on the Riemannian manifolds, we can obtain existence and nonexistence of solution.
|
PDF Full Text Request