Asymptotic Behavior For Strongly Damped Wave Equation And Strongly Damped Finite Lattice System

The main theme of this paper is related to the asymptotic behavior of solutions for autonomous and non-autonomous strongly damped wave equations and discretized strongly damped difference equation (lattice system), considers that the existence of an attractor, estimating the dimension and structure.In Chapter 1, gives the background of dynamical system, and general introductions of researches of infinite dimensional dynamical systems and finite lattice systems. It introduces situations of the study of dynamical behavior of solutions for the damped wave equations and the main research results of this paper.In Chapter 2, we consider the dynamical behavior of the strongly damped wave equations under homogeneous Neumann boundary condition. Applying the spatial average method, by the property of limit set of asymptotic autonomous differential equations, we prove that in certain parameter region, the system has a one-dimensional global attractor, which is a horizontal curve.Chapter 3 considers the dynamical behavior of a second order strongly damped finite lattice system where the coupled operator is nonnegative definite symmetric. Firstly, we prove the existence of a global attractor, and give an upper bound of Hansdorff dimension of the global attractor, which keeps bounded for large strongly damping. Then we use the theory of restricted horizontal curve and rotation number to prove that when the damping term is linear and the strongly damping is suitable large, the system has an unbounded one-dimensional global attractor, which is a restricted horizontal curve.In Chapter 4, we study the asymptotic behavior of a uniform attractor for a strongly damped wave equations with time-dependent driving force. If the time-dependent function is translation compact, then in certain parameter region, the uniform attractor of the system has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the equation. And it attracts all the solutions of the equation with exponential rate. At the same time, in certain parameter region, we obtain an upper estimate for the Kolmogorov e-entropy and fractal dimension of the uniform attractor. Finally, we consider the strongly damped wave equations with rapidly oscillating external force g^ε(x, t) = g(x, t, t/ε) having the average g⁰(x, t) asε→0⁺. We prove that the Hausdorff distance between the uniform attractor. A_εof the original equation and the uniform attractor A₀ of the averaged equation is less than O(ε^1/2). We mention, in particular, that the obtained results can be used to study the usual damped wave equations.In Chapter 5, on the one hand, we consider the existence of global attractor of the strongly damped wave equations under homogeneous Neumann boundary condition. By the property of limit set of asymptotic time-periodic differential equations, we prove that in certain parameter region, the system has a one-dimensional global attractor, which is a horizontal curve. On the other hand, we discuss the existence of a global periodic attractor for the strongly damped nonlinear wave equations with non-autonomous time-periodic driving force under homogeneous Dirichlet boundary condition. By establishing first-order equation which is equivalent to these problem, and by means of introducing a new norm which is equivalent to the usual norm, it is proved that in certain parameter region, for arbitrary non-autonomous and time-periodic driving force, the strongly damped nonlinear wave equations of time-periodic driving force with homogeneous Dirichlet boundary conditions has a unique periodic solution, i.e., global periodic attractor, which attracts any bounded set exponentially.Chapter 6 studies the viscoelastic and thermoviscoelastic equations with homogeneous Dirichlet boundary conditions. It is shown that for arbitrary nonautonomous and time-periodic driving force, the system has a unique periodic solution attracting any bounded set exponentially, i.e. the global periodic attractor. And if the driving force is autonomous, the global periodic attractor is the unique equilibrium Solution which at- tracts any bounded set exponentially.