The Long Time Behavior Of The Suspension Bridge Equations With Linear Memory

In this master's dissertation,we mainly consider the long time behavior of the suspension bridge equations with linear memory.We apply the theories of infinite dimensional dynamical systems,combine with the energy estimate and the method of contraction function to obtain our main results.The specific content is as follows:In the first part,we introduce the development of infinite dimensional dynamical systems,as well as the research background and the current situation of the suspen-sion bridge equations.Moreover,we introduce the main problem and methods used in this paper.In the second part,we review some basic concepts of infinite dimensional dy-namical system and the abstract results of attractors which are prepared for the follow-up work.In the third part,we study the existence of strong global attractors for damped suspension bridge equations with past history.First,we obtain the existence of bounded absorbing sets in the strong space by using the skills of energy estimate,after that we overcome the difficulty about the verification of the compactness of semigroup,and then prove the existence of global attractor in the strong space.In the fourth part,we discuss the existence of uniform attractors for non-autonomous suspension bridge equations with nonlinear damping and past history.The presence of nonlinear damping term a?x?g?ut?makes the method of the proof is restricted,such as the method of operator decomposition to verify the compactness of semigroup is not completely applicable for dissipative evolution equations with nonlinear damping.Next,since the presence of the memory term,we can't use the test function?I-P_n?u to obtain the compactness of semigroup.Finally,we prove the main conclusion by applying the method of contraction function;moreover,we correct mistake about the estimate of semi-linear term u+ in[18,19].In the fifth part,we investigate the existence and regularity of global attractors for the extensible suspension bridge equations with past history.Due to the equation include the memory term and geometric nonlinearity(?-???u?_L²???²)?u,they bring some difficulties and complexities to verify the existence of bounded absorbing set and the compactness of semigroup.After more complex and carefully estimates,we prove the existence of global attractors making use of the method of contraction function.Furthermore,we obtain the optimal regularity of the global attractor by using the technique of operator decomposition.