【作者】 魏丽艳；

【导师】 刘亚成；

【作者基本信息】 哈尔滨工程大学 ， 应用数学， 2006， 硕士

【摘要】 本文研究以下强阻尼非线性波动方程的初边值问题其中Ω（?）Rⁿ为有界域。f∈C，且f（u）u≥0。首先，利用新定义的位势井族结合Galerkin方法对整体弱解的存在性进行研究，得到了新的弱解的存在条件。其次，利用一些重要的不等式如Holder不等式，Gronwall不等式等结合位势井族进一步对强解的存在性进行研究，得到了新的整体强解的存在条件及强解的唯一性。再次，在解的存在性基础上研究问题（1）-（3）在该族位势井的流之下的不变性，得到了解的真空隔离性质，即方程的所有解均在H₀¹（Ω）空间的一个小球的内部或一个大球的外部出现，而不会在中间的带形区域出现，形成一个无解区域称为真空隔离区域。最后，利用积分估计的方法研究了该问题解的渐近性质，并得到了较好的结果，使得解以指数形式趋于零。更多还原

【Abstract】 This paper deals with the initial boundary value problem of a class of strongly damped wave equationsWhere,Ω(?)R~n is a bounded domain and f∈C with f(u)u≥0.The existence of the weak solutions of the above problem is discussed by combining the Galerkin method and the family of new potential wells that defined in this paper, and the new existence terms are obtained. Besides, by utilyzing some important inequalities such as Holder inequality, Gronwall inequalities and the family of potetial wells, the existence of the strong solution of the problem is analyzed, and the new existence terms and uniqueness are gained then. Then, the invariance of the family of potential wells under the flow of (l)-(3) is reaserched, vacuum isolating property of the solutions is gained, that means all solutions of the equations may appear in the inside of a small ball or outside of a large ball, rather than the band-shape intervenient regine. Thus, a non-solution region called vacuum isolating region is formed. Finally, the asymptotic behaviors for solutions of the problem were studied by using integral estimate method. The rezults indicate that the solutions of the problems decay to zero according to the exponent of t.更多还原