A Class Of Nonlinear Diffusion Equations In Perforated Domains

In this monograph, we discuss a class of nonlinear diffusion equations in per-forated-like domains. Nonlinear diffusion equations, as an important class of parabolic equations, come from a variety of diffusion phenomena appeared widely in nature, see [5], [6], [7]. They are suggested as mathematical models of physical problems in many fields such as filtration, phase transition, image segmentation, biochemistry and dynamics of biological groups.Investigation about the partial differential equations in perforated domains can be seen a long time ago. As early as 1939, the former Soviet Union mathematician G. N. Bukharinov considered a functional equation in a perforated domain. Since that time, many mathematicians concentrated their minds on various kinds of differential equations in perforated domain, see for example [12]—[25]. The main characters of this class of problems are that all the problems were considered in the domain which can be regarded as the limit of digging out one or several small balls from a domain. And people's main interest is to study the asymptotic behavior of solutions when the radiuses of the small balls shrink to one point. Based on the background of the problems and the characteristics of the discussed equations, people can take various methods to deal with the idiographic problems. For example,people can use the asymptotic expansion of solutions, or apply the properties of the solutions of the limit equation, even can utilize directly the limit of the solution at the orifice which might be existent as the auxiliary condition to discuss the asymptotic behavior of the solutions. As we know, the method which by using the properties of the solutions for radially symmetric problems to discuss the properties of the solutions for partial differential equations is very important, see [26]-[30]. Since we have already acquiesced in that the domain we considered is the complementary set of digging some open balls from a whole domain, the centers of the balls can be regarded as the boundaries of the domain. As regards the source of the problems, the problems concerning the radially symmetric solutions of the partial differential equations can be considered as a class of extensive problems which are discussed on a perforated domain.When the limit of the solution is existent at the orifice, we mainly consider the asymptotic behavior of the solution. The problem which is studied in perforated domain is different from the problem in the domain without orifice. There is a close relationship and a remarkable difference between the two problems. For this class of problems, the solution of the problem in the domain without orifice, is always the solution of the problem in the corresponding perforated domain, but the contrary might be false. On one hand, for some proper boundary value condition at the orifice, perhaps the solution of the problem in the perforated domain can also solve the problem in the domain without orifice. On the other hand, it is a more important respect too, there may be such a situation, for any boundary value conditions at the orifice, all the solutions of the problem in the perforated domain can not solve the problem in the domain without orifice. As for the orifice, that means there are two possibility, the removable orifice and the unremovable orifice. The so-called unremovable orifice, namely refer to however the boundary value condition at the orifice is, the solution of the problem in the perforated domain can not solve theproblem in the domain without orifice. Otherwise, we call the orifice the removable orifice. We know that when the orifice is the removable orifice, people can solve the partial differential equation by studying its radially symmetric solutions. And when the orifice is the unremovable orifice, the problem is important too. Finding conditions for unremovable orifice is the main subject we want to investigate.This monograph is divided into three chapters.In the first chapter, we consider the following problem, that is the nonlinear diffusion equation with nonlocal boundary value condition in perforated domain,^- div {\x\a\Vu\p-2Vu) = f(x,u), (x,t)eRT, (1)\x\a\Vu\p~2Vu- n= [ |x|Q|Vw|p-2Vu ? Vgdx, {x, t) e dU x (0, T), (2)where RT = (^\{0}) x (0,T), fi\{0} is a perforated domain, 0 € fi, O C R" is a bounded domain with smooth boundary, n > 2, ri\{0} can be considered as the limit of Sl\Be, BE is a ball with radius small enough, a > 0, p > 2, n denotes the unit outward normal to the boundary dft. We call the condition (2) the nonlocal boundary value condition.Nonlocal boundary value problems of similar form were first considered by Bit-sadza [2], and later by Karakostas and Tsamatos, Cao Daomin and Ma Ruyun and IF in and Moiseev, etc., see [31]-[33]. Our considerations were motivated by the model considered by Karakostas and Tsamatos [1], in which they studied the following ordinary differential equationu"(r)+q(r)f(u(r),u'{r)) = 0,which corresponds to the special case n = 1, p — 2, a — 0 of the equation (1), with the nonlocal boundary value condition,u'(l) = / u'(s)dg(s), Jothe Dirichlet boundary value condition,lim u(r) = 0,r->0+and proved the existence of nonnegative solutions.The key of determining whether the origin is the unremovable orifice or not is how to present the boundary value condition at the origin, since suitable boundary value condition can make origin be the removable orifice. On one hand, we use [1] for reference. On the other hand, the derivative of the solution on origin may not exist either. Therefore in this monograph, we add Dirichlet boundary value condition at x = 0,lim u{x,t) = 0(t). (3)K)+We first consider the radial steady state, and the problem can be transformed intoVP(u'))' + r^MO/M = 0, re (0,1), (4) = f Jo n+a-x sn+a-xp{u\s))dg{s), (5)lim u(r) = 0, (6)r->0+where 6 is a constant. Here we use u(r) to represent the solution of the problem (4)-(6). We study the existence and uniqueness of solutions for the problem, and determine whether the origin is the removable orifice. We always assume n > 2, p > 2, a > 0. One can see that it is not a simple extension of [1], since the dimension n > 2, the exponent p ^ 2 and a > 0. In one respect, due to n > 2, a singular point r = 0 arises in the problem; as the exponent p^2, the problem is not only changed into a nonlinear problem, but also have degeneracy at the point where v! = 0; and for a > 0, degeneracy arises at the origin. On the other hand, the most important respect is that unlike the solutions in the whole domain ￡1, the radiallysymmetric solutions do not imply a natural condition u'(0) = 0 in general. In fact, for the solution u(r) what we look for, the derivative u'(r) might not exist at r = 0, and even unbounded in (0,1). We will show that there is a sharp threshold for the exponent p. Exactly speaking, if p > n + a, then the problem (4)-(6) have solutions. In fact, for any 9 > 0, the problem (4)-(6) admits one and only one solution. While if p < n + a, there is no solution of the problem (4)-(5). Utilizing the shooting method and Schauder's fixed point theorem, we prove the existence and uniqueness of the classical solution. Analyzing the order of growth near r — 0, we can see that when p < n + a, the classical solution is not existent. Also we can show the strict monotonicity of the solution with initial value condition. By the results of the radially symmetric problem, we present a exact condition. The condition can determine whether the origin is the removable orifice or not. And applying this condition, we can see that the origin is the unremovable orifice.Next, we consider the non-radial steady state. In order to conquer the degeneracy came from x = 0, we consider the problem in a domain with a hole, that is the domain which is made by digging out a small ball from the whole domain. For overcoming the degeneracy brought by \Vu\ = 0, we study the regularization of the elliptic equation and do the a priori estimates in a weighted space. By using Schauder's fixed point theorem, we prove the existence of weak solutions. We can select two radially symmetric problems, and let the two solutions of the two problems be the sub-solution and super-solution of the non-radial problem, then show that the origin is the unremovable orifice. Although the method in [1] can not be used to deal with the case with fixed degeneracy, the function at the right side of the equation can be of the more widely form f(u,u'). Though the equation considered in [1] is not of degeneracy, we can use the methods in [1] and apply the Krasnoselskii's fixed point theorem to deal with the case p > 2. However, we can only treat the one dimensional case.Finally, we will consider the more general form of the equation (1), the non-steady state. Using the similar methods to non-radial steady state, we can obtain that the weak solution satisfies the equation in the sense of distribution. We utilize the method based on sub-solutions and super-solutions to conquer the difficulties due to the degeneracy near x — 0, and then verify the boundary value condition at x — 0. Also using the method based on sub-solutions and super-solutions, we can conclude that the origin is the unremovable orifice.Through the research of the problem with nonlocal boundary value condition above, we find that the origin is the unremovable orifice. This made us naturally ponder over such a question, whether such a conclusion remains valid under any other boundary value conditions. However, it is not the case in general. Through the discussion on two kinds of important boundary value conditions, the Neumann boundary value condition and the Dirichlet boundary value condition, we find that different boundary value condition may have very tremendous influence to the problem. In the last two chapters of this monograph, we study the Neumann boundary value problem and the Dirichlet boundary value problem in perforated domain respectively.In the second chapter, we study the following Neumann boundary value problem in the domain Q\{0}:— - div (\x\a\Vur2Vu) = f(x,t), (x,t) g Rt, (7)\x\n\Vu|p~2Vw n= g(x, t), (x, t) G dQ x (0,T), (8)where a > 0, p > 1, n > 1. Similar to the case of nonlocal boundary value condition, we also consider the Dirichlet boundary value condition at x — 0,\\mu(x,t) = 6(t), te(0,T). (9)x+0The works dealt with the problems of Neumann boundary value conditions can be seen in [34]-[39]. And most authors considered the linear equations. For example, in [34], Doina Cioranescu and Andrey L. Piatnitski use eigenvalue method and multiple scale method to study the Neumann boundary value condition of the linear equationin a perforated domain, and obtain the asymptotic behavior of the solutions as the holes shrink to the point. In [35], Marc Briane apply a spectral approach to consider the Neumann boundary value condition for the linear equationand show the existence and the asymptotic behavior of solutions.Here we consider the nonlinear diffusion equation (7). First, we discuss the radial steady state. By analyzing the results of radially symmetric problem, we can gain the existence and uniqueness of solutions and present the accurate condition for the unremovable orifice and the removable orifice at the origin. We know, generally speaking, when the derivative of the solution at the origin is zero, the origin is usually the removable orifice. But to the radial problem, the conclusion that the derivative of the solution at the origin is zero does not true in general. In fact, by analyzing the results of radial steady state with the Neumann boundary value condition, we can find that many kinds of situations are all contingent. For example, perhaps the derivative of the solution at the origin is zero, it is also possible that the derivative of the solution at the origin is existent but is not zero, or even the derivative is not existent. We will also discuss the case of non-radial steady state and non-steady state separately. Similar to the method mentioned in the first chapter, we use a series of methods, such as doing the a priori estimates in a weighted space, establish the comparison principle, etc., to gain the existence and uniqueness of the solutionfor the problem. Although we can obtain the condition both of the unremovable orifice and the removable orifice on origin in the case of radial steady state, we encounter the difficulties on nonradial steady state and nonsteady state. Finally, we can only use the method based on sub-solutions and super-solutions to show that the origin is the unremovable orifice. It is also our pity that we can not obtain the condition of the removable orifice on origin, when the case in nonradial steady state and nonsteady state.In the final chapter, we discuss the problem with the Dirichlet boundary value condition in the perforated domainu(x,t) = g{x,t), xedQx{Q,T), (10)and we also consider the Dirichlet boundary value condition (9) at x = 0. More early works of studying the linear equations with the Dirichlet boundary value condition in perforated domain can be seen in [20].Afterwards, the former Soviet Union mathematician O. A. Oleinik, A. S. Shamaev and G. A. Iosifyan carried on their research to the linear equationAu = f(x),with the Dirichlet boundary value condition in perforated domain and obtain the asymptotic behavior of the solution. Our consideration is the nonlinear equation. We will discuss the steady state of equation (7) emphatically, namely, consider the following problem- div (\x\Q\Vu\p-2Vu) = /(;r), x € Q\{0}, (11)u(x) = g(x), x ￡ m, (12)limu(x) = e. (13)x)0...