Some Properties Of The Solutions Of Nonlinear Diffusion Equations

It is well-known that solutions of the heat equation have infinite propagation of perturbation. This means that solutions are positive for any t > 0 if the initial data are nonnegative and non-trivial, see the monographs [1], [2]. Obviously, regardless of mathematics or physical angle, such a kind of phenomenon can't be accepted. This indicated that the heat propagation, as a mathematical model, cannot reflect the real situation accurately and has the flaw essentially.It is discovered that the nonlinear heat equation may overcome the insufficiency which the linear heat equation brought. The nonlinear heat equation model even more approaches in the actual situation. In fact, the solutions of nonlinear heat equation with slow diffusion have finite propagation of perturbation, for example, the p-Laplacian equation with p > 2, and the porous medium equation with m > 1. Namely, if the initial data u₀ has compact support, the solutions u(·,t) have also compact supports for the space variable x. See [3], [4], [5], [6], [7], [8]. On the other hand, the extinction property characterize the solutions for nonlinear heat equation with fast diffusion, for example, the p-Laplacian equation with 1 < p < 2, and the porous medium equation with m < 1. Namely, there exists some finite time T(u₀) such that u(·,T(u₀)) = 0. See [9], [10], [11], [12], [13], [14], [15], [16], [17]. Essentially the nonlinearity results in the interesting phenomenon .In Chapter 1, we deal with the the obstruction phenomenon of the solutions of the evolutionary p-Laplacian equation with the source term f. Here, the source term is focused on a nonempty subdomain. That is to say, letω(?)Ωbe an nonemptydomain, then f∈L^q(Ω×(0, T)) andf≡0 a.e. inΩ\(?)×(0, T).we call f satisfying the properties above a local source term. Such kind of research work for the evolutionary p-Laplacian equation with local source term does not still much been found up to now.By virtue of the De Giorgi technique, we obtain the following results:For any local source term f, the local L¹-norm and L^∞-norm of the solutions to the evolutionary p-Laplacian equation are uniformly bounded (with respect to f) to the fast diffusion case (i.e. 1 < p < 2) and the slow diffusion case (i.e. p > 2) respectively.It have been investigated widely by many authors the extinction phenomenon of linear and nonliear heat equations. For example, Diaz([9]), Lair[10], [21], Gaktionovand Vazquez [22], [23], Herro and Vazquez [6], DiBenedetto and Showalter [12] and so on have studied the Dirichlet problem for the porous medium equation with absorption term; Leung and Q. Zhang [11] , Nin Sun [24] have deal with the Neumann problem and the mixed-boundary value problem for the porous medium equation with absorption term; Zhao Junnin [25], Yuan Hongjuj and Gao Wenjie [26]have studied the extinction phenomenon for the Dirichlet problem of the evolutionaryp-Laplacian equation with absorption term. The common characteristic of these problems is that the studied equations have "the absorption item". Because the action of the absorption term, the system energy weakens along with the increase of time, Therefore, the solutions extinct in the finite time.It is worth pointing out that Y. Li and J.C.Wu [27] obtained the extinction phenomenon for fast diffusion porous medium equation with heat source term under some conditions.In Chapter 2, we deal with the initial-boundary problem for p-Laplacian equationwith heat source term. In virtue of the energy estimate and comparison principle,we obtain the global existence and the extinction phenomenon of the solutions, and the blow-up property of the solutions by the concave method. In Chapter 3, we generalize the results in [27] and Chapter 2 to the fast diffusion polypropic filtration equation. It is well known that the fast diffusion porous medium equation and the fast diffusion p-Laplacian equation has one singular point u = 0 and |â–½u| = 0 respectively. But the polypropic filtration equation has tow singular points u = 0 and |â–½u|= 0. We overcome the difficulties brought by the two singular points and obtained the similar results to ones in [27] and Chapter 2.In the last 20 years, the following systemhas been investigated extensively. Owing to its important and various applications in mathematical models of physical and biological problems in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. During the past two decade years, the nonlinear diffusion system above has attracted attentionof many authors. Kalashinikov, Wu Zuoqun, Yin Jingxue and Yuan Hongjun [40], [41], [42], [44] studied the existence and uniqueness of solution for the nonlineardiffusion system with filtration operator as its main part. In [40], Kalashinikov obtained the existence of the generalized solutions for the following system(i=1,2). But the uniqueness of solutions did not be obtained. Kalashinikov said in [41] the uniqueness of solution is an open problem. Yuan Hongjun partially answered the Kalashinikov's problem in [45].In Chapter 4, we deal with the existence and uniqueness of soultuions for the Cauchy problem of following evolutionary p-Laplacian system.with the inintial-boundary valueu_i(x,0)=u_0i(x),(?)_x∈R^N, wherem_i>2, p_i>0, q_i>0,a_i>0,b_i≥0are constants, and 0≤u_0i∈L¹(R^N)∩L^∞(R^N ),i=1,2.