Life Span Of Classical Solution Of Hyperbolic Geometric Flow

In this thesis, we study the lifespan of classical solutions to two kinds of hyperbolic geometry equations on high-dimensional manifolds.In Chapter 1, we introduce the background, current status of our study, and give the main results of our paper.In Chapter 2, we study the standard hyperbolic geometry flow on Riemann surfaces. Based on methods of constructing an approximate solution, character-istics and the blowup Lemma in [18], we show that the classical solutions of the hyperbolic geometry flow with small initial data depending only on r=(?) blow up in finite time. Moreover, we establish the explicit asymptotic expression of the lifespan Tε as ε→0.In Chapter 3, we concern the standard hyperbolic geometry flow equation in several space dimensions and derive the lower bound of lifespan of classical solutions of the hyperbolic geometry flow equation with "small" initial data by the standard method of continuous induction.In Chapter 4, we investigate the dissipative hyperbolic geometry flow on Riemann surfaces. A new nonlinear wave equation is derived. Based on the energy method, the global existence of the solution to the dissipative hyperbolic geometry flow is obtained. Moreover, under suitable assumptions, we reprove the global existence of classical solutions of the Cauchy problem, and show that the solution and its derivative decay to zero as t tends to infinity.Finally, we introduce the hyperbolic Yamabe problem in appendix A. In particular, we investigate the global existence and non-existence of smooth solu-tions of the hyperbolic Yamabe problem for the (1+n)-dimensional Minkowski space-time. More precisely speaking, for the case of n≤3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem and show that the flat (1+n)-dimensional Minkowski space-time can be conformal to a space-time with constant curvature. While for general multi- dimensional case n≥4, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.