| This thesis deals with the existence and regularities of the weak solutions of the initial-boundary value problem and the Cauchy problem for nonlinear degenerate parabolic equations. For the nonlinear degenerate parabolic equations in general form, we discuss its initial-boundary value problem in the anisotropic Sobolev space and get the existence result of the weak solution. The tool we mainly used is the compactness method. For two classes of quasilinear degenerate parabolic equations used extensively in application we consider their Cauchy problems and obtain the existence and regularities results of the weak solutions. Also, we give some examples to show the application of our regularity theorem. Here we mainly use the viscosity method and the maximum principle. The organization of this thesis is as follows.Chapter one considers the initial-boundary value problem for quasilinear degenerate parabolic equation in the general form as follows:O.A.Ladyzenskajia and N.N.Uralceva studied the above problem for non-degenerate equations in an ordinary Sobolev space in the 70's of the last century. Under the assumptions that the coefficient functions of the equation are monotone and integrable they got the existence of the weak solutions. As we know, a few papers deal with the strongly degenerate equations. But in the real world, there are many infiltration and conductive problems in the mixed medium, also since the sudden change of some physical properties in mixed medium it is necessary to study their mathematical model, i.e., the quasilinear degenerate parabolic equations in the anisotropic Sobolev space that is the start point of our Chapter one. Counterpart to O.A.Ladyzenskaja's conditions we give the different integrability in different directions and the strongly degenerate condi- tions. We then generalize their works from non-degenerate problem in isotropic Sobolev space to the degenerate problem in the anisotropic Sobolev space. Our result includes some results of the later works after O.A.Ladyzenskajia's work. The main result in this Chapter is as follows:Theorem 1.2.1 If ai(x,t,u,p) and a(x,t,u,p) satisfy the conditions (A1), (A2) and (A3), then for any ψ0∈L2(Ω), problem (1.1.1) has at least a solution in Vm,2(QT).This work has been published in Acta Mathematica Scientia (SCI), 2006,26 B(2):255-264.).In Chapter two, we study the Cauchy problem for the quasilinear degenerate parabolic equation as followswhich has the strongly physical background and studied by lot of mathematician. For the existence of the weak solution there was a rough proof in [22] and in this thesis we give a intense argument for the existence. The main contribution in this Chapter is our regularity theorem. As earlier as the year 1990, M.Bertsch, D.Passo and M.Ughi proved that for the different dimensions of the space and the different ranges of the parameter 7 the regularity results are as follows: is not necessaryly continuous in(Ω|-).In 2000, Lu Yunguang and Qian Liwen generalized the above regularity results:if then the weak solution u ia Lipschitz continuous in x and H(o|¨)lder continuous with the exponent 1/2.Hence, we see that the regularity of the viscosity solution of problem(2.1.1) depends on the dimension number N and the parameter 7. The main contribution of this Chapter is to generalize the regularity result from r ≥ (2N)1/2ï¼1 to r ≥ (Nï¼1)1/2 (obviously,(Nï¼1)1/2<(2N)1/2ï¼1). In this way, we generalized all of the results obtained by previous authors to a new one. As we know, our result is the best one up to now. The main result in this Chapter are as follows:After proving the existence of the the viscosity solution of the problem we haveTheorem 2.2.1([24]) The viscosity solution of problem(2.1.1) is just the weak solution of problem(2.1.1).Theorem 2.3.1 If u is the viscosity solution of problem (2.1.1), r≥(Nï¼1)1/2 (N ≠10) and there exist constantssuch that . Then, in RN × [0, T], the viscosity solution u is Lipschitz continuous in x and Holder continuous in t with the exponent 1/2.This regularity result has been accepted for publication by Nonlinear Analysis TMA (SCI), you can find it in the website of this journal by searching the authors Yan-Yan Zhao and Zu-Chi Chen).In Chapter three we still consider the Cauchy problem but for a more extensive and significant quasilinear degenerate parabolic equations:It is necessary to start this Chapter since many significant degenerate problems are beyond Chapter two. We got the existence and regularity of the weak solution to this problem, i.e., Theorem 3.2.1 If β2 = β1ï¼1,β1 ≥ 0,α1(β1ï¼1)ï¼Î±2>0, α1(β1ï¼ 1)ï¼2α2>0, then the viscosity solution u of problem (3.1.1)is just the weak solution of problem (3.1.1).Theorem 3.3.2 Suppose that α1, α2, β1, β2, u0 satisfy the conditions in theorem 3.2.1, and there exists a non-positive constant s ≠-2 such that2α2β1ï¼2α2ï¼sα2+2s(s+1)α1+ Nα1β12 ≤0.Then, in the region of RN × [0, T], the viscosity solution u(x,t) of problem (3.1.1) is Lipschitz continuous in x and Holder continuous in t with the exponent 1/2.In some sense this Chapter is the generalization of Chapter two. Although we still use the viscosity method and the maximum principle to prove the existence and the regularity of the solution but this problem is more complicated than the problem in Chapter two ant it requires much supper techniques. Also, the result we got in this Chapter including many significant results, especially the one in Chapter two. At the same time we proved independently the existence of the viscosity solution to this problem. Therefore, this Chapter is the further development of Chapter two.The regularity result in this Chapter has been published in Electronic Journal of Differential Equations, 2007, No.15,1-6.
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