Study On The Global Dynamical Behavior For Several Kinds Of Nonlinear Solid Structure System

Beam, plate and shell structures are the basic and key load-bearing components of engineering. The stability directly determines their service time in the complex environment. Due to its theoratical and practical significance, it is highly desirable to study the long-time dynamical behavior and the dynamical stability of the components.In practice, most elastic components are close to the continuum of mechanical systems. Under the effect of the potential force field, the study on the stability of continuous systems involves nonlinear partial differential equations, dynamical systems with infinite degree of freedom, infinite dimensional space, etc. Analysis of the stability of nonlinear elastic systems based on dynamics is thus a focus of research.Prior works have covered the existence, the uniqueness and the asymptotic dynamical behaviors of the solutions to elastic beam and plate functions. However, little attention has been devoted to the long-time dynamical behaviors. The attractor is an important indicator of the long-time dynamical behavior as time tends to infinity. The geometrical characteristics of the attractor is measured by the fractal dimension. The existence and the estimation of the dimensions of the attractor have become another hot topic of infinite dimensional dynamical systems.In this thesis, the solid structures with strong damping, structural damping, or external damping are investigated. First, we analyzed long-time dynamical behaviors of the solutions to a class of non-autonomous beam systems with strong damping and external damping that follow Dirichlet boundary conditions, utt+αΔ2u+μΔ2ut+γu-m(|▽u|2)Δu-n(t)Δut+f(ut)= g(x,t). By utilizing the operator semigroup theorem,the uniqueness of the continuous solutions has been proved. Additionally, the semigroups of operators of autonomous systems are generalized to non-autonomous systems by introducing an equivalent norm.Given certain conditions,the bounded absorbing set using the uniform priori estimates of the energy can be obtained.Via the process decomposition technique,the process is decomposed to two fractions,in which one fraction satisfies the squeezing property and the other satisfies the uniformly compact property. As the result, we proved the existence of the compact Kernel sections of the corresponding process,and further proved the existence of the uniform attractor of the generating process of the systems.Second, we studied the long-time dynamical behaviors of the solutions to the following Kirchhoff type coupled thermoelastic beam systems with strong damping and structural damping, utt+Δ2u+αΔ2ut-[M(|▽u|2)+N(∫Ω▽ut·▽udx)]Δu+βΔθ=f(x) θt-Δθ-βΔut=g(x) By using the operator semigroup theorem, we proved there exists an unique mild solution with given bounded coefficients. We also constructed equivalent functional based on the operator semigroup theorem. Using the bounded absorbing set that is obtained from the uniform priori estimates of the energy, we proved the existence of the global attractor of the semigroupsFinally,we studied the long-time dynamical behaviors of the solutions to the following thermoelastic plate systems with strong damping, utt+Δ2u-γΔut+αΔθ+N1(u)N2(θ)=f1(x,t) [θt-βΔθ-αΔut+N3(u)N4(θ)=f2(x,t) We proved the uniqueness of the continuous solutions of the systems by employing the operator semigroup theorem.We obtained the bounded absorbing set of the semigroups by introducing an equivalent norm,energy methods and a series of fine priori estimates.Then we proved the existence of the global attractor of the semigroups.At last,we estimated the Hausdorff dimension of the attractor through variational methods and the uniform priori estimates of the energy.