Parabolic Geometric Partial Differential Equations Of The Overall Solution

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...