【作者】 黄建宁；

【导师】 段志文；

【作者基本信息】 华中科技大学 ， 应用数学， 2009， 硕士

【摘要】 本文主要考虑n维紧黎曼流形(M,g)上的抛物Monge-Ampere方程：整体解的存在性与非存在性,其中λ和p都是实参数且p>1,-f,φ0：M→(0,+∞)都是流形M上的光滑函数,g是流形M上的黎曼度量.文章分为三个部分：第一部分主要介绍先前和当前Monge-Ampere方程的研究动态和主要结果,该部分还包括本文所做的主要工作、获得的主要结论以及与流形相关部分知识的介绍；第二部分也就是正文部分,包括解的先验估计、n+△φ的估计以及本文主要结果的证明：若λ>0,则方程(*)的整体解φ(x,t)存在,且当t→∞时,解φt=φ(·,t)收敛到其定态方程的唯一解φ∞(x)；第三部分主要考虑λ<0且f=0的情形,应用抛物方程的比较原理,获得了两个结果：(ⅰ)若方程的初值φ0(x)>0,则方程(*)的整体解不存在；(ⅱ)方程(*)的定态方程：不存在大于零的解φ∞(x).更多还原

【Abstract】 In this paper, we just consider the existence and non-existence of global solutions for parabolic Monge-Ampere equations on compact Riemannian Manifolds (M,g).Here A and p> 1 are real parameters.-f/,φ0:M→(0,+∞) are smooth functions on compact Riemannian manifold (M,g) and g is a Riemannian metric on M. This article is divided into three parts:In the first part, we just introduce previous and current situation of Monge-Ampere equations as well as main results that have acquired in their papers. Of course, the work done by this paper and main results that have acquired in this article are included in this part. Prepared knowledge of Riemannian Manifolds are also introduced in the first part. The body of this article is the second part. In the second part, it includes a priori estimates of solution and n+Δφ. The main conclusions of this paper are as follows:ifλ> 0, then the solutionφof equation (*) exists for all times t andφt=φ(·,t) converges exponentially towards a solutionφ∞of its stationary equation as t→∞. In the third part, supposingλ< 0 and f= 0, two results will be obtained by applying Comparison Principle of parabolic equations to equation (*):First, ifφ0(x)> 0, then the global solution of equation (*) doesn’t exist; Second, the stationary equation of equation (*):does not have positive solutionφ∞(x).更多还原