Classical Solutions Of Two Classes Of Hyperbolic Equations

The study on hyperbolic equations is an important topic in the theory of partial dif-ferential equations,occupying an important position in the field of mathematics research today.Hyperbolic equations can describe vibrations or wave phenomenon in nature,engineering and technology.In 1746,in his paper?Recherches des courbes forme parvi-bration de la corde tendue?,D 'Alembert derived the string vibration equation firstly,and gave the solution to the Cauchy problem for the string vibration equation.The study of string vibration pioneered the subjeet of partial differential equations.So far,the theory on hyperbolic equations or wave equations has developed very well,but most are concerned with small solution.For the Cauchy problems with periodic or big initial data,the corresponding theory is not complete yet.This thesis studies the classical solutions of two classes of hyperbolic equations:one is the elastieity equations,which can deseribe the motion of elastie materials;the other is the hyperbolic mean curvature flow equations,which can govern the evolution of plane curves.We will consider the periodic solutions and self-similar solutions,respectively.The thesis has the following three chapters:the first chapter firstly introduces the research background and the present situation of hyperbolic equations,and it also introduces the main results of this paper and the strueture arrangement.In the second chapter,by introducing the Riemann invariants,we can reduce the equations of elasticity to quasilinear hyperbolic equations,and then obtain the blow up of periodic solutions for the elasticity equations and the estimation of the life span.The third chapter studies the self-similar solutions of the hyperbolic mean curvature flow.We prove that all curves immersed in the plane which move in a self-similar manner under the hyperbolic mean curvature flow are straight lines and circles.Moreover,it is found that a circle can either expand to a larger one and then converge to a point,or shrink directly and converge to a point,where the curvature approaches to infinity.