【作者】 李娜；

【导师】 张建文；

【作者基本信息】 太原理工大学 ， 数学， 2023， 硕士

【摘要】 在非线性发展方程中,色散和耗散是完全不同的两种机制,这两种机制对研究此类模型解的适定性和解的长时间动力学行为有着重要的研究意义.而且,梁是日常生活中最常见的一种建筑材料,在实际应用中,往往需要考虑梁的热弹性.因此本文以弹性梁中波导为对象,考虑了在固体结构中最常见的粘性耗散性质,非线性源项和色散效应等因素,研究两类广义双色散热弹耦合梁方程系统的初边值问题,建立了其解的适定性和长时间动力行为.具体内容如下:第一章:介绍本文研究问题的研究背景和研究现状,并给出本文接下来的主要工作.第二章:引入本文在研究过程中所用到的基本概念,引理,定理和常用不等式.第三章:研究了一类广义双色散热弹耦合梁方程组在边界条件u=△u=θ=0,（x,t）∈（?）Ω×R⁺.和初始条件u（x,0）=u₀（x）,u_t（x,0）=u₁（x）,θ（x,0）=θ₀（x）,x∈Ω.下的初边值问题,其中（x,t）∈Ω×R⁺,Ω是Rⁿ中具有光滑边界（?）Ω的有界域,a,b,γ,ω∈（0,1）以及α为正常数.首先通过Faedo-Galerkin方法证明了整体解的适定性;然后基于上述整体解的存在唯一性定义了动力系统（H,S（t））,最后通过构造合适的能量泛函和Lyapunov函数证明系统的衰减性和渐近紧性,验证了该系统在齐次边界条件下全局吸引子的存在性.第四章:考虑内部反馈中带有时滞项和热记忆项的一类广义双色散热弹耦合梁方程组的长时间动力学行为.与前一章相比,本章证明出系统整体解具有更强的正则性以及根据定理2.2.4证出该系统在空间H_-δ中存在广义指数吸引子.第五章:对本文进行简单的总结,并作出下一步的研究计划.更多还原

【Abstract】 In the nonlinear evolution equations,dispersion and dissipation are two completely dif-ferent mechanisms,which play a significant role in studying the well-posedness and the long time dynamic behavior of the solutions of such models.Moreover,beam is the most common building material in daily life.In practical application,it is often necessary to consider the thermoelasticity of beam.Therefore,this paper takes the waveguide in the elastic beam as the object,and considers the most common properties of viscous dissipation,nonlinear source term and dispersion effect in solid structures.And then we study the initial boundary value problem of two kinds of the generalized double dispersion thermoelastic coupled beam Sys-tems to establish the well-posedness and the long time behavior of their solutions.The specific content is as follows:The first chapter introduces the research background and the research status of this paper,and gives the following main work of this paper.The second chapter introduces the basic definitons,lemmas and some common inequal-ities of studying the paper.The third chapter,we investigate a class of the generalized double dispersion thermoe-lastic coupled beam equationsunder the boundu=△u=θ=0,（x,t）∈（?）Ω×R⁺.and initial value conditionu（x,0）=u₀（x）,u_t（x,0）=u₁（x）,θ（x,0）=θ₀（x）,x∈Ω.where（x,t）∈Ω×R⁺,Ωis a bounded domain with a smooth boundary（?）Ωin Rⁿ,a,b,γ,ω∈（0,1）andαare normal numbers.First of all,the well-posedness of the global solution is proved by the Faedo-Galerkin method,Then the dynamical system（H,S（t））is defined based on the existence and uniqueness of the above global solution,Finally,By constructing a suitable energy function and Lyapunov function,we prove the attenuation and asymptotic compactness of the system,and verify the existence of global attractors under homogeneous boundary conditions.The fourth chapter considers the generalized double dispersion thermoelastic beam e-quations with time delay and the thermal memoryCompared with the previous chapter,we prove that the global solution of the system has stronger regularity and the system has a generalized exponential attractor in the space H _-δaccording to the theorem 2.2.4,.The fifth chapter is a simple summary of this paper,and make the next research plan.更多还原