粘弹性方程-文献翻译

学校代码 11517 学 号 200911002104 HENAN INSTITUTE OF ENGINEERING 文献翻译 题 目 一类非线性粘弹性方程解的整体存在性 学生姓名 樊辰光 专业班级 信息与计算科学 0941 班 学 号 200911002104 系 部 理学院 指导教师 职称 李文清 讲师 完成时间 2013 年 5 月 20 日 0 统一衰减解一类边界消耗的非线性粘弹性波动方程统一衰减解一类边界消耗的非线性粘弹性波动方程 吴舜堂 1 1 介绍介绍 在本文中 我们所关心的是下面的粘弹性能量衰减率问题 LEM 具有非线性边界耗散 其中 和是一个有界域具有光滑边界 这里 和是关闭0 0 k 1 NRN 0 1 和不相交的 而且 v 是单元外向法线的 松弛函数 g 是一个积极的和均匀衰 减的功能 h 和 a 是满足一定条件的给定的 A2 和 A3 具体来说 Rb 是一个函数 而且和 uuuf p 2 通常这种类型的方程中出现的粘弹性的理论 这是众所周知的 粘弹性材料 具有记忆效应 这是由于由材料本身的历史的影响的力学响应 由于这些材料 具有广泛的应用在自然科学 他们的动力是有趣的和非常重要的 从数学的角 度来看 他们的记忆效果是模拟的积分方程 因此 有关的行为的 PDE 系统的 解决方案的问题已经引起了相当的关注 在最近几年 例如 卡瓦尔康蒂等 1 考虑了以下的问题 其中是有界域 N 1 用光滑的边界 Y 0 和一个 R 是一个函 N R 1 数 它是非零 作者建立了一个指数衰减估计的条件下 当 a x ao 0 时 且满足一些几何条件和 Berrimi 和 Messaoudi 2 改善的结果 1 通过引入一个新的功能 他们证明了 一个较弱的条件下 a 和 g 指数衰减结果 事实上 他们允许函数消失关于 任 何部分 并且 因此 边界的一部分上的几何形状所施加的条件 不再需要 随后 同一作者的 3 和 Messaoudi 4 一种情况 即一个源项与粘弹性耗散竞 争结果延长 在 5 卡瓦尔康蒂和奥肯多考虑以下因素 在一定条件下的松弛函数 g 他们改进的结果为 1 事实上 他们证明了该解 决方案 1 5 呈指数衰减到零 当 g 是呈指数衰减并且 h 是线性的 并将该溶 液多项式衰减到零 当 g 衰减多项式和 h 是非线性的 在考虑的边界稳定 卡 瓦尔康蒂等人 6 思考了后续问题 研究员假设了相当严的阻尼项 h 和内核函数 g 的存在性和一致衰减率结果 后 来 卡瓦尔康蒂等人认为这样的结果并不加强一个对 h 的生长条件 而是在 g 的条件下一个较弱的假设 最近 Messaoudi 和穆斯塔法 8 利用凸函数的一些 性质 9 和乘数方法来扩展这些结果 他们建立一个明确的和一般的衰变率 结 果没有施加任何限制增长的假设上的阻尼项 h 大大削弱的假设条件 近年来 问题 1 1 一直在被 Li 等人研究 10 用 b x 0 和函数 f u 1u1rU r 0 他们展现出了整体存在性和唯一性的全球问题的解决方案 1 1 在合适的条件 下 并建立了统一的能量的衰减率上的初始数据和松弛函数 g 我们阅读参考 相关作品 11 16 有处理边界稳定的问题 由以前的作品的启发 它的解决方案和统一的衰减结果有趣的是 调查 的整体存在性问题 1 1 当力源问题长期的和在假设 b 和条件较弱的粘弹性 耗散和非线性边界阻尼竞争时 事实上 我们将允许为空函数 包括 本身 的任何部分 和内核函数 g 不一定衰减的指数或多项式的方式 因此 我们的 结果允许更大范围内的类松弛函数 结果提高了在 10 13 其中只有对指数 2 和多项式率进行了审议 本文的其余部分安排如下 在第 2 节中 我们假设以后将用于提的地方在第 3 节的存在 resule 定理 2 1 我们证明了我们稳定的结果 给出定理 3 7 2 初步结果初步结果 在本节中 我们给出文中用到的的假设和预备知识 首先 我们介绍了以下一 组 和 与希尔伯特结构引起的 现有的这个是一个 Hilbert 空间 为简 单起见 我们记和根据 1 2 我们有结论 Sobolev 嵌入是最佳常数满足以下不等式 我们使用的跟踪 Sobolev 嵌入 在这种情况下 包埋 常数由 BL 表示 即 接下来 我们声明如下问题 1 1 的假设 是有界的功能满足 存在非增正可微函数 f A2 H R R 是一个非减函数 A3 a R 是一个非负的功能 例如 3 对一些积极的恒定的 a1 利用 Galerkin 方法和程序 10 16 类似 我们可以有以下的局部存在性问题 1 1 的结果 3 3 整体存在性和能量衰减 整体存在性和能量衰减 在本节中 我们把注意力集中统一的衰减问题 1 1 的弱解 为了这个目的 我们定义 和能量函数 当 采用 10 的证明 我们还有下面的结果 引理 3 1 对于任何我们都有 引理 3 2 设 a 是 1 1 的溶液中 然后 根据假设 A1 的一 A2 中 E t 的是一个 nonincreas ing 和 nction 在 0 T 接下来 我们定义的官能 F 这有助于在建立所期望的结果 设置 4 另外 从 3 1 3 2 2 4 和 F 的定义由式 3 6 我们有 引理 3 4 假设 A1 一 A2 和 1 2 接住 进一步假设 并满足然后 它认为 对于所有的 t E 0 T 有 对于所有的 t E 0 T 证明利用 3 8 并考虑 E T 是一个非增函数 我们得到 下一步 我们将证明 5 要建立 3 13 我们认为矛盾 假设 3 13 不成立 则存在 t E 0 T 使得 这与 3 11 和 3 11 这也是一个矛盾 因此 我们证明了不等式 3 13 为了证明 3 10 我们注意到例如 因此 我们完成了引理 3 4 的证明 参考文献参考文献 1 M M Cavalcanti V N Domingos Cavalcanti and J A Soriano Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping Electronic Journal of DentialEquations vol 44 pp 1 14 2002 2 S Berrimi and S A Messaoudi Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping Electronic Journal of Differential Equations vol 2004 no 88 pp 1 10 2004 3 S Berrimi and S A Messaoudi Existence and decay of solutions of a viscoelastic equation with a nonlinear source Nonlinear Analysis Theory Methods 6 Applications vol 64 no 10 pp 2314 2331 2006 4 S A Messaoudi General decay of the solution energy in a viscoelastic equation with a nonlinear source Nonlinear Analysis Theory Methods Applications vol 69 no 8 pp 2589 2598 2008 5 M M Cavalcanti and H P Oquendo Frictional versus viscoelastic damping in a semilinear wave equation SIAM Journal on Control and Optimization vol 42 no 4 pp 1310 1324 2003 6 M M Cavalcanti V N Domingos Cavalcanti J S Prates Filho and J A Soriano Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping Drential and Integral Equations vol 14 no 1 pp 85 116 2001 7 M M Cavalcanti V N Domingos Cavalcanti and P Martinez General decay rate estimates for viscoelastic dissipative systems Nonlinear Analysis Theory Methods Applications vol 68 no 1 pp 177 193 2008 Uniform Decay of Solutions for a Nonlinear Viscoelastic Wave Equation with Boundary Dissipation 7 Shun Tang Wu 1 Introduction In this paper we are concerned with the energy decay rate of the following viscoelastic problem with nonlinear boundary dissipation Where Ko 0 and is a bounded domain in R n 1 with a smooth boundary n R 10 Here and are closed and disjoint with meas I o 0 and v is the unit outward 0 1 normal to The relaxation function g is a positive and uniformly decaying function h and a are functions satisfying some conditions given in A2 and A3 respectively b R is a function and with uuuf p 2 This type of equations usually arise in the theory of viscoelasticity It is well known that viscoelastic materials have memory effects which is due to the mechanical response influenced by the history of the materials themselves As these materials have a wide application in the natural sciences their dynamics are interesting and of great importance From the mathematical point of view their memory effects are modeled by integrodifferential equations Hence questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years For example Cavalcanti et al 1 considered the following problem where is a bounded domain in R n 1 with a smooth boundary y 0 and 8 a R is a function which may be null on a part of The authors established an exponential decay estimate under the conditions that a x ao 0 and satisfying some geometry conditions and Berrimi and Messaoudi 2 improved the result 1 by introducing a new function They proved an exponential decay result under weaker conditions on both a and g In fact they allowed the function a to vanish on any part of and consequently the geometry condition imposed on a part of boundary is no longer needed Later the same authors 3 and Messaoudi 4 extended the result to a situation in which a source term is competing with the viscoelastic dissipation In 5 Cavalcanti and Oquendo considered the following Under some conditions on the relaxation function g they improved the result of 1 Indeed they proved that the solution of 1 5 decays exponentially to zero when g is decaying exponentially and h is linear and the solution decays polynomially to zero when g is decaying polynomially and h is nonlinear On considering the boundary stabilization Cavalcanti et al 6 considered the follow ing problem The existence and uniform decay rate results were established under quite restrictive assump dons on damping term h and the kernel function g Later Cavalcanti et al 7 generalized this result without imposing a growth condition on h and under a weaker assumption on g Recently Messaoudi and Mustafa 8 exploited some properties of convex functions 9 and the multiplier method to extend these results They established an explicit and general decay rate result without imposing any restrictive growth assumption on the damping term h and greatly weakened the assumption on fir Very recently problem 1 1 has been considered by Li et al 10 with b x 0 and f u 1u1rU r 0 They showed the global existence and uniqueness of global solution of problem 1 1 and established uniform decay rate of the energy under suitable conditions on the initial data and the relaxation function g We refer the 9 reader to related works 7 11 16 dealing with boundary stabilization Motivated by previous works it and uniform decay result of solutions is interesting to investigate the global existence to problem 1 1 when a forcing source term is competing with the viscoelastic dissipation and nonlinear boundary damping under the weaker assumption on both b and fir In fact we will allow the function b to be null on any part of including itself and the kernel function g is not necessarily decaying in an exponential or polynomial fashion Therefore our result allows a larger class of relaxation functions and improves the results in 10 13 where only the exponential and polynomial rate was considered The remainder of this paper is organized as follows In Section 2 we provide assumptions that will be used later and mention the local existence resule Theorem 2 1 In Section 3 we prove our stability result that is given in Theorem 3 7 2 Preliminary Results In this section we give assumptions and preliminaries that will be needed throughout the paper First we introduce the following set and endow with the Hilbert structure induced by We have that is a Hilbert space For simplicity we denote and According to 1 2 we have the imbedding be the optimal constant of Sobolev imbedding which satisfies the following inequality and we use the Trace Sobolev imbedding In this case the imbedding constant is denoted by Bl that is Next we state the assumptions for problem 1 1 as follows is a bounded Cl function satisfying 10 and there exists a nonincreasing positive differentiable function f such that a2 h R R is a nondecreasing function with A3 a R is a nonnegative functions such that for some positive constant al By using the Galerkin method and procedure similar to that of 10 16 we can have the following local existence result for problem 1 1 3 Global Existence and Energy Decay In this section we focus our attention on the uniform decay of weak solutions to problem 1 1 For this purpose we define and the energy function Where 11 Adopting the proof of 10 we still have the following results Lemma 3 1 For any we have Lemma 3 2 Let a be the solution of 1 1 then under assumptions A1 一 A2 E t is a nonincreas ing 和 nction on 0 T and Next we define a functional F which helps in establishing the desired results Setting Further from 3 1 3 2 2 4 and the definition of F by 3 6 we have Lemma 3 4 Suppose that A1 一 A2 and 1 2 hold Assume further that And satisfyThen it holds that 12 for all t E 0 T Moreover one has and for all t E 0 T Proof Using 3 8 and considering E t is a nonincreasing function we obtain Next we will prove that To establish 3 13 we argue by contradiction Suppose that 3 13 does not hold then there exists t E 0 T such that This contradicts 3 11 Then This is also a contradiction of 3 11 Thus we have proved the inequality 3 13 To prove 3 10 we note for such that 13 Therefore we complete the proof of Lemma 3 4 References 1 M M Cavalcanti V N Domingos Cavalcanti and J A Soriano Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping Electronic Journal of DentialEquations vol 44 pp 1 14 2002 2 S Berrimi and S A