Initial Boundary Value Problems And Cauchy Problem For A Class Of Nonlinear Wave Equation Of Higher Order

This paper consists of three chapters. The first chapter is introduction. In the second chapter, we will study the existence and uniqueness of the classical global solution and blow-up of solution to the initial boundary value problem for a class of nonlinear wave equation of higher order. In the third chapter, we will study the existence and uniqueness of the classical global solution and generalized global solution to the periodic boundary value problem and the Cauchy problem for this kind of equation.In the second chapter, we study the following nonlinear wave equation of higher order:with the initial boundary value conditions or with where a1 ,a2 ,a3 > 0 are constants, (s) ,f(s0 , s1 , s2 s3, s4 ) are given nonlin-ear functions, u0(x) and ,u1(x) are given initial functions. For this purpose, by Green's function of a boundary value problem for a fourth order ordinary differential equation we first reduce the problem ( 1) - (3) to an equivalent intergral equation, then making use of the contraction mapping principle we prove the existence and uniqueness of the local classical solution for the intergral equation. The existence and uniqueness of the classical global solution for this problem are also proved by means of the method of continuation of solution. The main results are the following:Theorem 1 Suppose that0, = 0,1 , and the following conditions are sati-where A , B > 0 areconstants,where FwhereThen the problem (1)-(3) has a unique dassical global solution. Theorem 2 Suppose that the following conditions hold:wheredenotes the norm of space L2(0,l),whereThen the classical solution of the problem (1) - (3) must blow up in finite time.As to the problems (1), (4), (5) and (1), (6), (7) ,we have the analogous results as Theorems 1 and 2 above.In the third chapter, we will study the Cauchy problemwhere a1,a2 ,a3 > 0 are constants,are given nonlin-ear functions, UQ(x),UI(X) are given initial functions. For this purpose, by means of Galerkin method we first prove the existence and uniqueness of the generalized global solution and the classical global solution for the following periodic boundary value problemThen by constructing the sequence of the periodic boundary value problems and taking limit, we will obtain the existence and uniqueness of the classical global solution to the Cauchy problem (8), (9). The results are the following: Theorem 3 Suppose that the following hold:whereThen the problem (10)-(12) has a unique classical global solution. Theorem 4 Suppose that the following hold:where F whereThen the Cauchy problem (8), (9) has a unique classical global solution.