Some Researches For Elliptic Boundary Value Problems On Singular Manifolds With Multiple Totally Characteristic Degenerate Directions And Evolution Equations

In this thesis, we study the elliptic boundary value problems on singular mani-folds with multiple totally characteristic degenerate directions, including the existence and the multiplicity of solutions, the existence and the multiplicity of sign-changing solutions; L2-regularity for solutions of the kinetic equation with external potential; the global existence and blow up of solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity. The full text is divided into six chapters. The details are as follows:In chapter 1, first we recall the history and the current situation for singular man-ifolds and elliptic boundary value problems on them (which are elliptic boundary value problems on singular manifolds with multiple totally characteristic degenerate direc-tions). Then we give an introduction from the beginning to the recent development of the problems for kinetic equations and pseudo-parabolic equations. In the last part, we present our main results.In chapter 2, we introduce the definitions of cone, edge, corner and related weighted Sobolev spaces on them respectively, as well as some fundamental inequalities. At the same time, we establish the distance function and Pohozaev identities on cone. We prove Hardy inequalities on corner. Finally, we summarize the generalized Hardy inequalities on singular manifolds.In chapter 3, we study the existence of positive solutions for the following asymp-totically linear cone-degenerate elliptic equation, ??B?????a(z)f(?),z ?RN+. Using Pohozaev identity, the barycenter function and some properties of solutions for the limit equation, as well as the Linking theorem, we obtain a nontrivial positive solution of this equation.In chapter 4, we consider the existence and the multiplicity of solutions and sign- changing solutions for the semilinear edge-degenerate elliptic equation, Here ?>0, N=1+n+q, n?1, q? 1,2? p?2N/N-2. Applying the symmetric Mountain Pass Lemma and a new Linking Theorem, we prove the existence of infinitely many solutions and sign-changing solutions respectively.In chapter 5, we study first the spectral decomposition for the corner-degenerate Laplace operator with Hardy function (which is-?Mu-?Vu). Then applying the results above, we get the existence and the multiplicity of solutions and sign-changing solutions for the following degenerate elliptic equation with potential term, At last, by using a perturbation method, we obtain the existence and the multiplicity of sign-changing solutions for the following perturbed semilinear elliptic equation,In chapter 6, we establish the L2 regularity for solutions of the following kinetic equation with external potential, (?)tu+y·?xu-?xV(x)·?yu= f(t,x,y). Making use of the multiplier method, we get the regularity for spatial variables from the regularity of velocity variables. Here we get rid of some restrictions on the index in Bouchut's paper [J]. Math. Pure Appl.,81,2002] and obtain more generalized results. At the same time, we estimate the external potential. Finally, applying the potential well method, we obtain the global existence, the asymptotic estimate and blow up at +? for solutions of the following pseudo-parabolic equation with logarithmic nonlinearity, Our result is different from the result in Xu-Su's paper [J. Punct. Anal.,264,2013]. They get that the solution blows up in finite time when the right hand side of the above equation is a polynomial nonlinear term. Meanwhile, we can only obtain that the solution grows in an arbitrary polynomial order due to the impact of the viscosity term Aut.