Initial Boundary Value Problem For Two Classes Of Nonlinear Evolution Equations

This paper consists of four Chapters.The first chapter is the introduction. In the second chapter,we will study the existence and uniqueness of the local generalized solution to the initial boundary value problem for a class of nonlinear wave equation of higher order.In the third chapter,we will prove the blow up of the solution to the problem mentioned in Chapter two. In the fourth chapter, we will discuss the existence of the global generalized and classical solution to the initial boundary value problem for the three dimensional Ginzburg-landau model equation, prove the solution tends to zero in L2 space as t approaches to infinity, and give the sufficient conditions of blow up of the solution.In the second chapter, we will study the following nonlinear wave equation of higher orderwith the boundary value conditionsand the initial conditionor with the boundary value conditionsand the initial condition (3) or with the boundary value conditionsand the initial condition (3), where k1 > 0, k2 > 0 are constants, g(s) is given nonlinear function, f(x, t) is a given function involving of x and t, (x) and (x) are given initial functions, V is the gradient operator, Rn(n = 1,2,3) is a bounded domain with sufficiently smooth boundary , v is the outward normal to the boundary , QT = (0, T). For this purpose, we first consider the following linear equationwhere G(x, t) is given. When the existence and uniqueness of the generalized solution to the problem (6), (2), (3) or the problem (6), (4), (3) or the problem (6), (5), (3) are proved, by making use of the contraction mapping principle we prove the existence and uniqueness of the local generalized solution to the problem (1), (2), (3) or the problem (1), (4), (3) or the problem (1), (5), (3). The main results are the following:Theorem 1 Suppose that . Ifholds, then there exists a unique local generalized solution u(x, t) to the problem (1), (2), (3) or the problem (1), (4), (3) or the problem (1), (5), (3) and , where [0,T0) is the maximal time interval.In Chapter three, the blow up of the local generalized solution to the problem (1), (2), (3) or the problem (1), (4), (3) or the problem (1), (5), (3) are proved by means of the concavity method. The main results are the following:where 0 > 0 is a constant, | | is the measure of n, for the problem (1), (4), (3), the condition g(Q) - 0 is satisfied;Then the solution of the problem (1), (2), (3) or the problem (1), (4), (3) or the problem (1), (5), (3) blows up in finite time if one of the following conditions holds:Theorem 3 Under the conditions (H1), (H2), and (H3) in Theorem 2, the solution of the problem (1), (2). (3) or the problem (1), (4), (3) or the problem (1), (5), (3) blows up in finite time if one of thefollowing assumptions holds:In the fourth chapter, we will discuss the following initial boundary value problemwhere a1 > 0, a > 0, a2 0 are physical constants,V is he gradient operator, g(s) and G(s) are given nonlinear functions, uo(X) is a known initial function, ft C R3 is a bounded domain with a enough smooth boundary , T > 0 is an arbitrary real number, QT = x (0, T). In the process of proving, we mainly use the Galerkin's method and Leray- Schauder's fixed point theorem. The main results are as follows:Theorem 4 Supose that the following conditions hold:(a) g(s) = s3, G C2(R), Vs 6 R, G'(s) , where 0 is a constant, | G(s) | d | s |p, 10 is a constant;(b) (s) = as3 + a2s, (s) G(s) 0;Then the problem (7)-(9) has a unique global solution u(x, t)holds.Theorem 5 If the conditions of Theorem 4 hold, and G C5(R), G"(0) = 0, u0 H6( ). Then the problem (7)-(9) has a unique global generalized solution.Theorem 6 Suppose that the following conditions hold:(6) G C1(R). G(0) = 0, there exists a constant b > 0 and s R, G'(s) -b;Then the genealized or classical solution u(x, t) to the problem (7)-(9) has the asymptotic property:Theorem 7 Suppose uo H2(n), 9 C3(R) and G C1(R), then the problem (7)-(9) has a unique gen...