抛物型几何偏微分方程的整体解

【作者】 汝强；

【导师】 李骏； 孔德兴；

【作者基本信息】 浙江大学 ， 基础数学， 2013， 博士

【摘要】 本文研究了定义在黎曼流形上的抛物型偏微分方程整体解的存在性、不存在性及其渐进性。论文由以下四章组成：第一章为绪论部分。在这一章中我们回顾了反应扩散方程组和Monge-Ampere方程的研究历史及其一些重要的结果,并叙述了本文的主要结果。第二章研究了定义在非紧致完备黎曼流形上的反应扩散方程组正整体解的存在性和不存在性,并获得了Fujita?型临界指标。第三章研究了一类定义在紧致完备的黎曼流形上的抛物型Monge-Ampere方程解的渐进行为,通过能量估计和Gronwall方法,得到了对应于B.Huisken文献[]中渐进性更为一般化的结论。第四章我们引入一类具有旋转不变量的新的几何流,证明了其整体解的存在性,并研究了在这种特殊情况下,沿着这一几何流,随着时间t趋向于无穷大,欧式空间Rn+1(n≥1)中任一封闭收缩光滑的超曲面会在C∞拓扑意义下趋向于一个圆。更多还原

【Abstract】 This thesis concerns three kinds of geometric partial differential equations defined on Riemannian manifolds. We investigate the existence of global exis-tence solution, the nonexistence of global solution and asymptotic behavior of solution of these equations. This thesis is organized as follows:In Chapter1, we recall some research history and well known results on reaction-diffusion system, Monge-Ampere equation and state the main results obtained in this thesis.In Chapter2, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction-diffusion system where Mn (n≥3) is a non-compact complete Riemannian manifold, and obtain a critical exponent of Fujita type.Chapter3is devoted to the asymptotic behavior of the parabolic Monge-Ampere equation on a compact complete Riemannian manifold. By using the Gronwall inequality and the energy estimates, we obtains asymptotic result of the corresponding problem which generalizes B.Huisken’s convergence result in []In Chapter4, we introduce a new geometric flow with rotational invariance and prove the existence of global solution.we also show that in a special case, under this kind of flow, an arbitrary smooth closed contractible hypersurface in the Euclidean space Rn+1(n≥1) converges to Sn in the C∞-topology as l goes to the infinity更多还原