Existance And Uniqueness Of The Entire Solution To Canchy Problem Of Evolution Equation

In many important subjects .such as physics and dynamic .basic equations are all partial differential equations .Movement of any substance subject to certain natural routine .Usual partial differential equations are mathematical model describing movement of relation between some physical value which is determined by corresponding physical law.Conservation of mass.conservation of moment muni and conservation of energy are basic regularity obeyed by all movement in nature.For a certain problem in natrue .if we quantify the corresponding conversation law . we can deduce partial differential equation reflecting this problem .and every differential equation reflecting a special physical phenomenon on certain codition.For example .wave equation . in the form of followingdescribes many kinds of physical process .When it is one dimension , it is chord vibration equation known to us.Moreover.it also describes' vibration of drumhead and spread of electromagnet wave.Another example is heat-conduction equation .in the form of followingit can describe heat conduction in even medium.thickness of diffusive substance and diffusion of artical.ect.Both of these example are evolution equations.Evlution equation is the equation which describes physical value changing with time.When we discuss evolution equation .firstly.we consider wether solution of equation exist.In discussion of solution of partial differential equation.we always consider it's solution on certain condition of deciding solution.For the evolution equation which has bounded domain.both initial condition and boundary condition are given.this kind of problem of deciding solution is called initial boundary .provlem. For evolution equation in whole space.initial condition is given.This kind of problem of deciding solutionis called Cauchy problem. Problem of deciding solution whose boundary condition is given is called boundary problem.In this thesis.Cauchy problem of evolution equation is discussed.For most equations in the form of the above, their solutions can not be worked out easily.So we concern ourselves with wether the two kinds of equation exist.However.practical problems are always complexed and most physical phenomenon are not on ideal condition.If we persist in thinking of their classical solution.our thought are limited and we think some equations which have solutions as without solution.For mathematical perfection and coformity to objectivity.we discuss solution in the wide sense in the Soblev space. About Soblev space .everyone can cansult reference[6].This thesis is divided into two parts.In one part.existenece and uniquess of global solution of wave equation is discussed. Existence and uniqueness of global solution of parabolic equations is discussed in the other part.