Hyperbolic Geometry Solution Of Partial Differential Equations

In this article, we study the global solution of the hyperbolic geometric partial differential equations. The paper is organized as follow:First, in the first chapter, we briefly summarize predecessors’work on the hyperbolic geometric partial differential equations.In the second chapter, for semi-linear wave equations with null form non-linearities on R3+1, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future.We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.In the third chapter, in M2+1, consider the wave maps with S2as the target, we show that for all positive numbers To>0and E0>0, there exist Cauchy initial data with energy at least E0, so that the solution’s life-span is at least [0,T0]. We assume neither symmetry nor closeness to harmonic maps.In the forth chapter, we investigate the Einstein’s hyperbolic geometric flow which is a natural tool to deform the shape of a manifold and to understand the wave character of metrics and the wave phenomenon of curvatures for evolution-ary manifolds. For the initial metric being both Einstein and totally umbilical, we prove that the solution metric is Einstein if and only if the solution mani-fold is a totally umbilical hypersurface in the induced space-time. For the initial metric being both Einstein and totally umbilical and possessing a constant mean curvature, we prove that the solution metric preserves to be Einstein, and the solution manifold is a totally umbilical hypersurface in the induced space-time, moreover the global existence and blowup phenomenon of the solution metric are also investigated. In the fifth chapter, we investigate group-invariant solutions of the hyper-bolic geometric flow on Riemann surfaces, which include solutions of separation variables, traveling wave solutions, self-similar solutions and radial solutions. In the proceeding of reduction, there are elliptic, hyperbolic and mixed types of equations. For the first kind of equation, some exact solutions are found; while for the last two kinds, with implicit solutions found, we furthermore investigate weather there will be a global solution or blowing up. Referring to Kong, Liu and Xu [28], the results come out perfectly.