| Considering the effect of Fourier or Gurtin-Pipkin conductivity law, we studied several classes of the thermoelstic coupled beam equations system.We obtained the existence and uniqueness of the global solution of the system under the homogeneous boundary condition, nonhomogeneous boundary condition and the initial value condition, and the existence of the global attractor for the system with homogeneous boundary condition and the initial value condition. The details of the full text is as follows:The first chapter briefly introduced the research of the thermoelastic coupled beam equations at home and abroad and summeried the main contents of this paper.The second chapter introduced the basic definitons, lemmas and some common inequalities of studying the paper.The third chapter investigated that the existence and uniqueness of the global solution for the non-autonomous thermoelastic coupled beam equations system with rotational inertial term and structural and nonlinear external damping, which is effected by Fourier conductivity law under the homogeneous boundary condition u= Δu=θ= 0 x∈(?)Ω,t∈Rτ,t≥τ and initial value condition by using Faedo-Galerkin method, with some skills of inequalities and prior estimate, which Ω(?) C Rn is bounded region with fully sufficiently smooth boundary condition of Rn, and 0<ω< 1, v> 0.The fourth chapter mainly studied the initial boundary value problem of the ther-moelastic coupled beam equations with nonlinear external damping under the nonlinear boundary conditions with certain initial condition u (x,0)= u0 (x), ut (x,0)= u1 (x),θ (x,0)=θ0 (x) x∈E (0,l)The fifth chapter proved the existence of the global attractor of the extensible beam equation with thermal memory term Firstly, we obtained the existence and uniqueness of the global solution for the system by using semigroup operator methods and sobolev space theory, with some skills of inequalities and prior estimations; Secondly, we definited a dynamical system ((?), S(t)) under the existence and uniqueness of the global solution. Then we proved the existence of absorbing sets and the dissipation of the system by using some skills of inequalities; Thirdly,we prove the system is asymptotically compact, which means the existence of the global attractors, by constructing some Lyapunov functions. |
