Asymptotic Behavior Of Solutions For The Suspension Bridge Equations With Nonlinear Damping And External Influence

Suspension bridge means a bridge takes the cable(or steel chain),which is hanged by tower and anchored on both sides(or both ends of the bridge),as an upper structure of the main load-bearing elements.Compared to other bridge structures,suspension bridge can use less material to span longer distance.Suspension bridge is suitable for the valley,rivers and other natural barrier regions.Due to these advantages,the construction method of suspension bridge is mostly used in modern bridges.At the same time,the security issues of suspension bridge have also attracted the attention of many scholars.Thus constructing a reasonable mathematical model and investigating the physical properties of suspension bridge are extremely necessary.In particular,it is of great practical significance and application value to construct the suspension bridge equations and to study the existence,decay and blow-up properties of the solutions for these equations.In this work,we investigate the asymptotic behavior of solutions of suspension bridge equations with nonlinear damping and external force.The thesis is organized as follows:In the first Chapter,we review the background and some development of suspension bridge problems and briefly describe the main work of the present thesis.In Chapter 2,we study the global existence,decay and blow-up of solutions for a suspen-sion bridge equation with nonlinear damping term |ut|m-2ut and source term |u|p-2u.Firstly,we establish the existence and uniqueness of local solutions by using the contraction mapping principle.Then,we give necessary and sufficient condition for global existence and energy decay results without considering the relation between nm and p.The proof of global existence result is based on the potential well theory and the continuation principle;while for energy decay result,the proof is based on the Nakao's inequality.Moreover,when p>m,we give sufficient condition for finite time blow-up of solutions.At last,by using an auxiliary functional,we establish the lower bound of the blow-up time Tmax.In Chapter 3,we investigate a suspension bridge problem with nonlinear damping term g1(ut(x,y,t)),nonlinear time delay damping term g2(ut(x,y,t-?))as well as nonlinear source term h(u(x,y,t)).We prove the existence of global weak solutions as well as a global attractor.We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure.Then we prove that this system processes a global attractor.We divide our proof into two parts:Firstly,we prove the existence of a bounded absorbing set;Then,we prove the asymptotic smoothness of the semigroup S(t).In Chapter 4.we summarize the contents of this paper,introduce the innowition of this paper,and list the contents that we can continue to study in the future.