Uniqueness Of Weak Solutions Of A Class Of Degenerate Parabolic Equation

Nonlinear diffusion equations, as an important class of parabolic equations, have profound background, and they are the mathematical formulation of diffusion phenomena appeared widely in nature. They are involved in many mathematical or physical fields of science, such as filtration and dynamics of biological groups and so on. Among these equations, there are two kinds which are most elementary but quite important, i.e. the following Newtonian filtration equations with the typical exampleand the non-Newtonian filtration equations with the typical exampleThe character of both equations is their degeneracy. Comparing to linear equations and quasilinear equations without degeneracy and singularity, such nonlinear diffusion equations with degeneracy reflect even more exactly the physical reality. So, they attracted numerous mathematicians' attention both in China and abroad. They took up with the researches on theory and application of this classof equations, including existence, nonexistence, uniqueness, asymptotic property and Blow-up etc. There have been a lot of corresponding literature dealing with the results. See for example [1, 10-13, 15, 16],Filtration is a kind of common phenomenon in nature, which indicates the movement of liquid in porous media. For example, the water flowing among the soil is a kind of filtration. The research of filtration is very important to the exploitation of ground water resouces and the discovery of petroleum or gas, especially to the agricultrue. At the same time, when we investigate the problem about the saline-alkali soil and melioration, the using fertilizer intelligently, the industrial waste water disposal and the protection of the ground water resource, which are involved the solute movemnet and the heat transfer, we must consider the dynamics of the solute in the filtration and the heat transportation.The experimental research of filtration phenomenon originated from the famous experiment of H. Dary's [1] in 1956. In many years after that, a lot of mathematical models were established, and researches on numerical computation and the theoretical qualitative analysis have been achieved a great deal progress. In this paper, we are instrested in the so-called polypropic filtration case. Suppose a compressible fluid flows in a homogeneous isotropic rigid porous medium. Then the volumetric moisture content 0, the seepage velocity V and the density of the fluid are governed by the continuityequatione^ + div(pv) = o,For non-Newtonian fluid, the linear Dracy's law is no longer valid, because the influence of many factors such as molecular and ion effects needs to be concerned. Instead, one has the following nonlinear relationpV = -AIVPI^VP,where pV and P denote the momentum velocity and pressure respectively, A > 0 and alpha > 0 are some physical constants. By transforming the varation and denotion we can get the so-called non-Newtonian filtration equation— = div(\Vu\p~2Vu).When we consider a flow with a homogeneous, isotropic and rigid porous medium filled with a fluid, Firstly, by the continuity equation, we have^ = 0, (1.1)where V denotes the macroscopic velocity of the fluid, 9 the volumetric moistrue content. The Darcy's law yieldsV = -fc(0)V$, (1.2)where k{9) denotes the hydraulic conductivity and $ the total potential. If we ignore the absorption and chmical, osmotic and ther-mal effects, $ can be expressed as$ = # + *, (1.3)where the first ^ is the hydrostatic potential due to capillary suction and z the gravitational potential. Here z is a variable which direction accords with gravitation.Combining (1.1), (1-2), (1.3), we obtain! (1.4)For many medium, \& could be a funtion of 0, i.e.^ = ^(0), Then we have the following equation of the formpig^ = AA(9)-{-dwB(9). (1.5)And the experiment yields that the hydraulic conductivity k{6) is not negative, i.e. A{9) is a non-decreasing function. This is a kind of typical filtration equation.On the other hand, if 9 depends on \￡, i.e. 9 = 0(^1/), the equation(1.4) is induced to39In one dimensional case, we could get the following equation by some proper transformIf the effect of gravitation is ignored, the equation (1.6) has the formdC{u) _ d2uwhere C{u) is in general a non-decreasing funtion. This equation is applied to the research of filtration with saturated and unsaturated region.For the equations (1.5), (1.6), (1.7) of above, we are interested in the degenerate. Generally, equation (1.5) is the typical parabolic-hyperbolic mixed equation, which degenerates when A'(s) = 0. While equations (1.6), (1.7) are elliptic-parabolic mixed types, which degenerates when C'(s) = 0.The theory on the solutions of the degenrate equations (1.5) couled ascend to 1958. In this year, Oleinik, Kalashinkov and Zhou Yulin [2] studied the Cauchy problem of the equations with thefollowing formdu d2ift(x,t,u)~di= dx~2 "Where they required tp(x, t, u) is defined for u > 0 and with the following propertiesi/j(x, t, u) > 0, t/)u(x, t, u) > 0, ^u > Ofcf, ip(x,t,Q) =ipu(x,t,0) = 0.Due to the degeneracy, classical solutions may not exist. They put forward the definitions of the generalized solutions of the firstboundary value condition and the second boundary value condition. Using the method of parabolic regularization, they proved the existence of generalized solutions. And they also proved the uniqueness of solutions and obtained the conditions for solutions to have the properties of finite propagation of disturbances.After that, Gilding and Peletier [3] considered the Cauchy problem for the equationdu _ d2um dyJ^ dt dx2 dxwhere m > 1, n > 0, and proved that it admis at most one generalized solution whenever n > -~{m -f- 1), and it admits a generalized solution if ?o nongenative, bounded and continuous with it,? lying Lipschitz continuous. Soon afterwards these results were extended by Gilding [4] to more general equationsdu d ( ,du\ ,dua = &r?SJ+J(%' (L8)where a(u), b(u) are continuous, and a(u) > 0(u > 0), a(0) = 0, He got the conclusion that there was at most one soltution of the equation whenb\u) = O(a{u)) (u -> 0+)Later, Chen Yazhe removed any kind of conditons for (1.8) with bin) controlled by a(u) and established the uniqueness by inforcing conditions only on a(u). While a substaintial progress in the studyof uniqueness was made by Prof. Zhao Junning [6] , who considered the equationdu d dt dxa(x, t, u)— J + Trb(x, t, u) + c(x, i, u),where a(x, t, u) > 0, and the set F = {s, a(s) = 0} does not include any interior point;when considering the first boundary problem, it doesn't need to be limited to zero boudary. Furthermore, he did not require a{u) and b(u) have any relation, and the uniqueness was porved in L°°(QT).Besides, in [7], the authors investigated another kind of weakly degenerate parabolic equationF( (19)dt dx V dx J dx ' [ }with F'(s) > 0, A'(s) > 0. Since degeneracy happens either when A'(s) — 0 or when F'(s) — 0, such kind of equations are called double degenerate. In particular, it turns out to be the Newtonian equations when A(s) = sm(m > 1), B(s) — 0 and the non-Newtonian equation when F{s) = \s\p~2s, A{s) = sm{m > 1), B(s) = 0. When F(s) = s, the equation(1.9) turn out to bedu _ d*A(u) dB(u)dt" dx2 + dx ' l }where A{s), B(s) G Cl(R), A'{s) > 0, but the set E = {s;A'(s) = 0} does not include any interior point.It was Kalashnikov [8] who first studied■a special case of (1.9). For this kind equation, there are more technical difficulty for the existence and uniqueness of the solution. Which comes from the strong nonlinearity in the equations. Kalashnikov accomplished the existence of the continuous solution for the Cauchy problem under the convexity conditions on the nonlinear functions A(s) and F(s).The convexity condition which Kalashnikov assumed implies that both Ea = {s;A'{s) = 0} and Ep = {s;F(s) = 0} contain at most one point. Namely, the equation considered has at most one point of degeneracy. Prof. Yin Jingxue [9] removed the convexity condition and accomplished a limit process for approximate solutions to obtain a continuous solution by means of parabolic reg-ularization, using the technique of BV estimates. What he consider is the boundary value problem for equations (1.9) with convection term. To prove the existence, the only condition islim F{s) = ±oo (1.12)s—>±oowhich is necessary in order to obtain continuous solutions. An investigation of Bertsch and Dal Passo [7] shows that even for a special case of (1.11), namely, for the equationut =the solution might be discontinuous if the condition (1.12) is removed. In addtion, Prof. Yin Jingxue [9] proved the uniqueness of continous BV solutions of the first boundary value for (1.9) with any stucture condtion.We will investigate the equation of the form)dB(u) ^H------—, (x, t) G Qt-, (1-13)where QT = (0,1) x (0, T), andA(s) = / a(a)da, B{s) = / b(a)da, Jo Jowhere a(a) > 0, b(a) is appropriately smooth. Especially, when a(s) has the form thata(cr) = (p-l)\ap-2, A fdu\ we have A — =du dxequation. Assume the condition as followsp-2du dx. Then (1.13) is the famous p-Laplacianboundary value condition and initial valueu(0,t) = u(l,t) = 0, u(x,0) = uo(x),t > 0,xe (0,1).(1.14) (1.15)For the equations (1.13)—(1.15) of above, we are interested in the degenerate case. That is the equation (1.13) with p > 2.We will study the uniqueness of weak solutions of the first boundary value problem for equation (1.13). Due to the degeneracy,the solutions may not be classical, so we formulate an approximate definition of this problem. By means of Holmgren's approach, we convert the uniqueness of the weak solutions to some estimates of solutions of the adjoint problem.