Properties Of Extinction And Shrinking Of Supports For Solutions To Nonlinear Reaction-diffusion Equations

Diffusion equations, as an important class of parabolic equations, come from a variety of diffusion phenomena appeared widely in nature, they are suggested as mathematical models of physical problem in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. In the last four decades, the study in this direction attracts large number of mathematician both in China and abroad. Remarkable progress has been achieved. Under the proper initial-boundary conditions, many authors have given considerable investigations on the solutions to equations, especially, the singularities of solutions which caused by the nonlinearities of diffusion terms, absorption terms, convection terms, and various coupling among them. In this thesis, we will give some qualitative analysis, such as extinction in finite time, instantaneous shrinking of support and the global existence of free boundaries, for several diffusion equations arose in applied sciences.In Chapter 2, we will investigate the extinction of solutions to several initial-boundary problems of evolution p-Laplacian equation. For the diffusion equation with homogeneous Dirichlet boundary condition. By using energy methods and modified comparison principle, we will give sufficient and necessary conditions of the extinction of solutions. We will show that, in the case of fast diffusion, if diffusion is stronger than source and the initial data is small enough, the solutions of problem will become extinct in finite time. Moreover, during the proof, we will obtain the upper estimate of the extinction time. For the problem with nonlocal source, we will establish local existence of solution. At last, we will consider the Cauchy problem of evolution p-Laplace equation in R~N. At the aid of comparison principle and constructing super-and subsolutions, we prove the sufficient and necessary conditions of the extinction of solution. When Harnack inequality is failed, we will give a critical exponent, which shows the effect of the decay behaviors of initial data at large distance on the extinc- tion of solutions.In Chapter 3, we will consider the Neumann problems of several parabolic equations with nonlocal sources and absorption terms. At first, for the problem with interior absorption and boundary source, by using modified comparison principle, we will improve the known results on sufficient condition of the extinction of solutions, and prove that, in the case of strong source, the phenomena of extinction should occur in finite time, also. Next, with the absorption in the interior domain or on the boundary respectively, by comparing with the standard results on blow-up of solutions, on the one hand, for the case of interior absorption, we will see, if absorption is stronger than interior source, the solution may vanish in finite time. On the other hand, for the case of boundary absorption, according to the definition of extinction, by using testing function methods, we will prove the solution cannot vanish in finite time. While the boundary absorption is stronger than the sources on the boundary, the solution may vanish on the boundary in finite time.In Chapter 4, we consider a parabolic equation with convection term and varia-tional coefficient and investigate the property of instantaneous shrinking of support of solution, that is, at any positive time, the support of solution is compact, irrespective of the support of initial data. This class of nonlinear diffusion equation with convection term resemble to the celebrated equation arising in statistical mechanics and have important physical sense. At first, we will prove the local existence and uniqueness of the solution, and establish two comparison lemmas. Then, at the aid of comparison lemmas, we will prove the property of instantaneous shrinking of support depends on the behaviors of variational coefficient.In Chapter 5, we will consider a nonlinear diffusion equation with absorption term and variational coefficient. This equation describes the diffusion coefficient depends on a quantity and its gradient in a diffusion procedure. It's a generality of the porous medium equation and evolution p-Laplace equation. The variational coefficient which depends on space variable, new properties appear. Applying energy methods and a iteration lemma convergence of sequences of numbers, we will prove the global existence of free boundaries and give some estimates to them.