Existence Of Solutions To A Class Of Nonlinear Parabolic Equations With Double Degeneracy

There exist, some diffusion phenomena in many fields such as filtration, bio chemistry and dynamics of biological groups. The research on these phenomena can be transformed into the study of nonlinear degenerate parabolic equations in divergence or not in divergence form. At present many authors focus on the research of the theory of degenerate parabolic equations about the local existence of weak solutions with special conditions, however few discussed with generally circumstances.In this paper, we are concerned with the initial and boundary value problem for the following nonlinear degenerate parabolic equations, that is The initial and boundary value is where p＞1,γ＞0,σ≥1,Ωis a bounded domain with appropriately smooth boundary (?)Ωin R^N.Ω_T=Ω×(0, T), In above equations Each one of f(x, t, u) has different definition, we discuss in the following chapters respectively. Since all equations not in divergence form degenerate whenever u=0 orâ–½u=0, the problems don't possess classical solutions in general. Therefore, we need to consider their weak solutions. Notice that these problems are related to the Newtonian filtration equations (p=2) and the Non-Newtonian filtration equations which have clearly physical significance forγ=0 or f(x, t, u)=0. If m(p-1)-1≠0 and let Hence v satisfiesThen equation (5) can be transformed formally into equation (1). The corresponding relations are as follow:γ＞0 corresponding to m＞1／(p-1),i.e. p＞1+1／m(slow diffusion);γ＜0 corresponding to m＜1／(p-1),i.e. p＜1+1／m(fast diffusion);γ=0 corresponding to the limit case when m→∞.For the classical positive solutions, (1) is equivalent to (5), and any one of them can be derived from the other by simple calculation. However, this degenerate equations have no classical solutions in general, even they have, the classical solutions are not necessarily positive everywhere. Hence, (1) and (5) only have the corresponding relation formally.Ifσ∈(0,1), f(x, t, u)=0, letting Then (5) can be transformed formally into (3). Problem (3) arises in biological and astrophysical context. Similar problems arise in some models describing physical phenomena. As for the above circumstances, we discuss all equations under generally conditions in this paper, that is the existence of weak solutions if f(x, t, u) > 0. Most of the results of (1) and (3) obtained as f(x, t, u)=0 and p=2, now the equation only degenerate at u=0. It was A. S. Kalashnikov who first studied the equation for p=2, N=1 andγ＞0; he proved the uniqueness of weak solutions in a certain kind of function. Many authors discussed the equation (3) when p=2,σ＞0 and f(x,t,u)=0. J. Hulshof and J. L. Vazquez studied the asymptotic behavior of the solution obtained by the viscosity vanishing method. M. Winkler discussed the qualities of solutions to the equations. Other related results can be found in the references therein. Few talked about the equations as p≠2. M. Tsutsumi studied weak solutions of the equation (3) in which p＞1,σ＜1 and f(x,t,u)=0, now the equation can be transformed into the following form:At present, W. Zhou and Z. Wu discussed the existence of weak solutions as p＞2 and constructed the multi solutions in the sense of distribution. W. Zhou and Z. Wu discussed some qualities of the equations (1) such as the existence and uniqueness etc. as p＞2,γ∈(0,1). Z. Yao and W. Zhou proved that the existence and uniqueness of weak solutions when p＞1 andσ＞1, and some other qualities such as blow-up also proved. For the case p=2, the discussion of the uniqueness of solutions can be carried out since the Laplace Operator possess some particular qualities. For the p—Laplacian, however, those convenient qualities will disappear. To overcome those difficulties, we have to redefine the solutions. As seen below, by combining the special construction of, we present the different definition of the solutions. Although it seems not to be natural and a little complex, the existence and uniqueness of weak solutions is guaranteed.The paper will be divided into three chapters.In the first chapter, we mainly discuss the following initial and boundary value problem of degenerate equation (A) not in divergence form where is a positive constant, And u₀ satisfies:Since the equation with double degeneracy is not in divergence form, the classical solutions may not exist. When f(x, t, u) satisfies some Conditions, we apply the method of parabolic regularization to prove the existence of weak solutions to the problem. Since the equation has double degeneracy, we make the regularization to both the initial and boundary value and the equation corresponding to the two different degeneracy. In order to obtain weak solution, we need to proceed two limiting process. By carefully analysis we make a sequence of estimates to the solutions, prove the weak convergence of the approximation solution sequence and hence obtain the existence of weak solutions. The results obtained generalize original results and also provided useful method in further study. The main theorem of the chapter is as follow:Theorem 1 Let p≥2,γ∈(0, 1), the problem (A) exists a weak solution u with the hypothesis (S), (u^μ)_t∈L²(Ω_T), whereμ=γp/2(p-1)+1/2.In the second chapter, we discuss the following initial and boundary value problem of the nonlinear degenerate parabolic equation It is difficult to prove estimations of the approximate solution sequence since the equation has u^σ-1. At last, under proper conditions, we obtain the weak convergence of the solution sequence applying the parabolic regularization method to get a series of estimates. Surmounting the difficult of the increasing function, we get the existence of weak solutions. This provide new profitable train of thought to further study. The main theorem is:Theorem 2 Suppose p≥2,σ∈[1, 2),γ∈(0, 1), the equation(B) has a weak solution u with the. hypothesis (H), (u^μ)_t∈L²(Ω_T), whereμ=γp/2(p-1)-σ/2+1.In the third chapter, the initial and boundary.value problem of the nonlinear degenerate parabolic equation with positive nonhomogeneous will be discussedwhereσ≥1, p＞1, f(x, t, u):Ω×(0, T)×R→R, f(x,t,u) is a positive continue differential function, and f(x, t, u) is monotone with respect to u, f(x, t, u)=(?)F(x,u)/(?)u,F(x, u) is differential with u. At the same time, u₀ satisfies the following conditions:Here f(x, t, u)＞0, it is more difficult to estimate weak solution sequence. By regularizing the initial and boundary value and the equation respectively, we get a series of estimations of the approximate weak solution. We have to possess two limits processes to pledge the weak convergence to get the existence and uniqueness. Finally, we derive blow-up of weak solutions in finite time by the classical concave method. The main theorems isTheorem 3 (Local existence) Let p＞1,σ≥1 and assume u0 satisfies (H). Then there exists a positive constant T^*=T^*(u₀) such that problem (C) inΩ_T^* admits a unique local solution.Theorem 4 (Blow-up) Let p＞1,σ∈[1, 2). Suppose u₀ satisfies 0≤u₀∈L^∞(Ω)∩W₀^1,p(Ω), and Then u(x, t; u₀) blow up in finite time.