【作者】 王艳萍；

【导师】 陈国旺；

【作者基本信息】 郑州大学 ， 基础数学， 2003， 博士

【摘要】 本文讨论几类非线性发展方程的初边值问题，在一定条件下证明这些问题解的整体存在性、唯一性，并给出解发生爆破的充分条件；同时还研究两类高阶非线性发展方程的时间周期问题，证明问题解的存在唯一性或弱解的存在性，主要结果包括下面五部分内容． 在第二章中，讨论如下的广义双耗散方程的初边值问题用Galerkin方法证明问题的整体广义解和整体古典解的存在性与唯一性，并通过建立一个新的不等式证明整体解不存在的充分条件．并利用得到的结果进一步讨论双耗散方程(DDE)和立方双耗散方程(CDDE)具有初边值条件(2)，(3)的初边值问题．得到的主要结果为： 定理1 假定且f′(s)是下有界的，则问题(1)-(3)存在唯一的整体广义解这里Ω=(0，e)．是具有充分光滑边界撇的有界区域.利用位势井方法结合方法证明问题小初值条件下的整体广义解的存在性与唯一性，并给出解爆破的充分条件，得到的主要结果是:证明问题(12)-(14)解的存在性与唯一性.其中可x，约是未知函数，F(、)和G(5)是给定的非线性函数，f(x，约是关于t以。>0为周期的已知函数，是常数.得到的主要结果是: 定理10假定下面的条件成立典解，并且如果M充分小，古典解是唯一的.在第六章中，讨论如下一类高阶非线性波动方程的时间周期问题利用临界点理论证明问题(15)-(17)弱解的存在性，其中是常数.得到的主要结果是:定理12如果。>1，则问题(15升(17)存在时间周期弱解更多还原

【Abstract】 In this paper, we study the existence and uniqueness of global solutions for the initial boundary value problems to some classes of nonlinear evolution equations, and give the sufficient conditions of blowup of solutions for the above problems, we also discuss the time-periodic problems for two classes of nonlinear evolution equations of higher order and prove the existence and uniqueness of time-periodic solution or the existence of weak time- periodic solution. The main results include the following five aspects:In Chapter 2, we study the following initial boundary value problem for a generalized double dispersive equationThe existence and uniqueness of the generalized global solution and the classical global solution are proved by Galerkin method. Then the sufficient conditions of the nonexis-tence of global solution are given by constructing a new inequality. Moreover, we also consider the following double dispersion equation (DDE)and the general cubic DDE(CDDE)with the initial boundary value conditions (2),(3) respectively. The main results are stated as follows:has a unique global generalized solutionbounded below.generalized solution or a classical solution of theSuppose that the following conditions are satisfied:converges when d > 0, moreover, B < I; the integralhas a unique global generalized solution ube the generalized solution of the problemSuppose that the following conditions are satisfied:for some finite time t0In Chapter 3, we investigate the initial boundary value problem for the wave equation with nonlinear damping and source termswhich describes nonlinear vibration of elastic rods. The existence and uniqueness of the global solution are proved in case that the initial data are small by potential well method combined with Galerkin method, we also give the sufficient conditions of blowup of the solutions for the problem (6)-(8) in finite time.The main results are stated as follows:Theorem 6 Assume that the following conditions hold.here K4 > 0 and , > 0 are constants. Then the problem (6)-(8) has a unique global generalized soluition u(t, x) which satisfiesIn Chapter 4, we discuss the initial boundary value problem for a class of nonlinear wave equationwhich describes the water wave with surface tension. The existence and uniqueness of generalized local solution are proved by Galerkin method and compactness argument and the sufficient conditions of blow-up of the generalized solution are given, by concavity method.The main results are stated as follows:the problem (9)-(ll) has a unique local generalized solution u(x,t). where ?Theorem 9 Assume is the generalized solution of the problem (9)-(l 1) and the following conditions hold:Jo where u(x) is the first normalized eigenfunction of the eigenvalue problemLet = be the corresponding first eigenvalue . Then there is a T T such thatIn chapter 5, we study the time- periodic problem to a generalized Ginzburg-Landau model equation in population problemsThe existence and uniqueness of the time-periodic generalized solution and the time-periodic classical solution to the problem (12)-(14) are proved by Galerkin method. The main results are the following two theorems: Then the problem (12)-(14) has a generalized time-periodic solutionTheorem 11 Assume the conditions of Theorem 10 and the following conditions are satisfied.has a classical time-periodic solution u(x,t}. Moreover, if M is sufficiently small, the classical solution is unique.In chapter 6, we consider the time-periodic problem for a class of nonlinear wave equation of higher orderBy means of the critical point theory, we prove the existence of the weakly time-periodic solution to the problem (15)-(17). The main result is as follows:更多还原