The Study Of The Cauchy Problem Of A Second Order Quasilinear Degenerate Parabolic Equation And The Parallelism Of A Riemannian Manifold

This paper is divided two chapters. The first chapter discusses the existence and uniqueness of solution for the Cauchy problem of the following quasilinear degenerate parabolic equationThe second chapter studies the geometric properties of a Riemannian manifold. We generalizes the concept of the parallel rays of Euclidian space to a complete noncompact Riemannian manifolds and discuss its properties. The above quasilinear degenerate parabolic equation arises from physics,mechanics, biology and finance etc fields. If aij =0,i,j = 1,2,...,n, the aboveequation is just the well-known conservation law equation, so there is not classical solution generally.When the space variable is one-dimension, mathematicians had made thoroughly study in the existence and uniqueness of Cauchy problem or the first boundary problem.When the space variables are multiple, the existence of the solution also had been thoroughly studied. But for the uniqueness of the solution, there are only a few results which study some special cases. In particular, there is not a satisfied result when the equation is strongly degenerate.By studying the known concepts and methods, this paper quotes a new concept of weak solution of the above equation. This concept contains some important information, which had been lost in the old definitions. On this newconcept, we succeed to prove the uniqueness of Cauchy Problem of the aboveequation by the methods of using the Hausdorff measure of the set of the discontinuous points of BV solution and Kruzkov technique etc. The existence of the solution we have defined is proved by regularized method and weakly convergent technique.In the second part of this paper, by considering the geometric properties of some well-known classical Riemannian manifold and combining with results we had got, we give the definition of the parallel rays in a complete nocompact Riemannian manifold M, which are usual parallel rays when M is restricted to Euclidean space R". By Toponogov Comparison Theorem, we prove that for agiven ray y in M which curvature is bounded below, there is a unique ray emanating from any point outside y such that the ray is parallel to r. If thesectional curvature of the manifold satisfies - a2 > KM > -b2, the parallel rayshave the same properties as in Euclidean space case. But if M with nonnegative curvature, it is essentially different from the Euclidean space case. Also we discuss the existence of infinite closed complete geodesic in a nocompact Riemannian manifold with nonnegative curvature.