【作者】 陈勇明；

【导师】 杨晗；

【作者基本信息】 西南交通大学 ， 应用数学， 2003， 硕士

【摘要】 对于一类非线性四阶波动方程的初边值问题：其中Ω(?)R~n为边界充分光滑的有界区域。 研究了其整体弱解的存在性、唯一性、光滑性和爆破性质。所得的四个主要结果如下： 1、运用Galerkin方法结合势井理论构造稳定集证明了： 定理(存在性)：设则问题(0.1)-(0.3)存在整体弱解u满足： 2、若p满足更强的条件时，运用能量方法结合不等式技巧证明了： 定理(唯一性)：设则(0.1)-(0.3)的整体弱解是唯一的。 3、运用Galerkin方法、稳定集和不等式技巧证明了： 定理(光滑性)：设则问题(0.1)-(0.3)存在唯一整体弱解西南交通大学硕士研究生学位论文第11页，，满足:。。L旬(o，T:H了(。)。万‘(。))u，。犷(0 .T:H了(卿)u，，任乙‘(0，了:L，(。))4、运用凸性分析方法结合势井理论构造不稳定集证明了:定理(.破):邵。。V，u，任LZ(。)，E(o)<d，u为问题(0.1)一(0.3)的局部解，则存在有限常数了，使得当l峥产时成立!。.2厂(。)范数意义下在有限时刻发生blow一uP.分二，即u在更多还原

【Abstract】 For the initial-boundary value problem of a kind of nonlinear fourth-order wave equations:where Rn is bounded domain with sufficiently smooth boundary.What studied in this paper are the global existence, uniqueness, smoothness and blow-up of the weak solutions of (0.1) - (0.3). Our four main results are stated as follows:1 By using the Galerkin method and constructing stable setaccording to the potential well theory, It is proved:Theorem (existence): Let . Then problem (0.1)-(0.3) has global weak solutions u satisfying:2 If p satisfies appropriately stronger condi tions, by using the energy method and the trick of inequality, It is proved:Theorem (uniqueness): Let . Then the global weak solutionof problem (0.1)-(0.3) is unique.3 By using the Galerkin method, stable set and trick of inequality, It is proved:Theorem (smoothness) : Let . Then the problem (0.1)-(0.3) has unique global weak solution u satisfying:4 By the convexity method and constructing unstable set according to the potential well theory, It is proved:Theorem (blow up): Let u, u is thelocal solution of problem (0.1) ?0.3), The there exists a finite constant T, such that. u blows up in f ini tetime under the L2() norm.更多还原