Research On The Global Well-Posedness Of Solutions For Some Dynamic Beam(Plate) Systems

The studies presented in the present thesis are from the research projects supported by the National Natural Science Foundation of China “The potential well theory and its applications to Kirchhoff System(11471087)”,“Qualitative study on nonlinear partial differential equations on singular manifolds(11871017)” and the Ph.D.Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities “The well-posedness of solutions to fourth order nonlinear damping wave equations(HEUGIP201808)”.This thesis devotes to the study of the dynamic behavior of the nonlinear dynamic suspension bridge system with nonlinear damping(dissipation),the nonlinear viscoelastic Euler-Bernoulli beam system with viscoelastic damping(dissipation)and frictional damping(nonlinear dissipation and strong dissipation),the fourth-order viscoelastic plate system with memory term and the thermal system(pseudo-parabolic)with nonlocal and strong dissipation terms by using potential well theory.This thesis focuses on the global well-posedness of the above systems,and respectively studies the initial conditions that lead to the global existence,asymptotic behavior and finite time blowup of solution,and the relationship between these initial conditions.We systematically study the dynamic behavior of a class of dynamic suspension bridge systems with nonlinear dissipation at three different initial energy levels.The subgrade of the dynamic suspension bridge is modeled by a rectangular thin plate.The two short sides of the thin plate are fixed on the ground,and the two long sides are free to oscillate,simulating the dynamics of the suspension bridge.The boundary of the thin plate,that is,the two fixed short sides and the two free long sides,is given by a specific boundary condition.The dynamic behavior of the suspension bridge system under the influence of external forces(including the gravity of the suspension bridge)can be simulated by the mathematical model established in this study.The global well-posedness of the solution for the mathematical model of the dynamic suspension bridge system is studied at low initial energy level,critical initial energy level and supercritical initial energy level.The solution of the mathematical model represents the displacement of the subgrade of the suspension bridge.This work focuses on the conditions for the global existence,the long time stability and the finite time blowup of solution to the mathematical model,which simulates the stability(the solution of mathematical model exists globally and decays)and instability(the solution blows up in finite time,i.e.,the suspension bridge collapses)of the subgrade of the suspension bridge at three different energy levels.This work systematically and structurally studies a class of nonlinear viscoelastic EulerBernoulli beam systems with viscoelastic damping and frictional damping(nonlinear weak damping and linear strong damping).Viscoelastic materials contain both the properties of purely elastic materials and purely viscous materials,such as amorphous polymers,semicrystalline polymers,biopolymers and even the living tissue and cells.This work focuses on the effects of viscoelastic damping and frictional damping on the well-posedness of the above nonlinear viscoelastic Euler-Bernoulli beam system.The existence of the damping term makes the structure of the system more complicated,so we must focus on overcoming the mathematical difficulties caused by the damping term in the research.Also the mathematical difficulties are mainly reflected in the study of the long time behavior of the global solution and the finite time blowup of the solution.Based on this,we introduce three auxiliary functions to prove the finite time blowup of the solution.Additionally,the exact estimates of the three damping terms are required in the whole analysis,and the estimates are different in different theorems.Our strategy divides the total initial energy of the system into three different initial energy levels by introducing the corresponding potential well depth.We first prove invariant manifolds of system with low initial energy level.Based on this,the global existence of solution is proved by the standard Galerkin method.Then we prove the energy decay exponentially by defining an auxiliary function in terms of energy with small disturbance due to strong damping.From the invariant manifold,the relationship between the potential well depth and some spatial norms of the solution is derived.The finite time blowup of solution of the system is proved at low initial energy level by introducing three suitable auxiliary functions.After that,by scaling the initial value,we extend all the results obtained at low initial energy level to the critical initial energy level in parallel.At supercritical initial energy level,we give some restrictions on initial energy and initial value in order to have the invariant manifold and prove the finite time blowup of solution.This work exactly studies the dynamic behavior of a class of fourth-order nonlinear viscoelastic Petrovsky systems on a rectangular thin plate,which can be used to simulate the dynamic properties of a viscoelastic plate.The subject focuses on the effects of viscoelastic damping and external forces on the dynamic behavior of rectangular plate.Firstly,the local existence of solution is proved by the classical Faedo-Galerkin method,the contraction mapping principle and the fixed point theorem.Then,the existence of global solution for system at subcritical initial energy level is obtained by combining the invariant manifold and continuity principle.The finite time blowup of solution at subcritical initial energy level is proved by introducing a family of potential wells and combining with the concave method.By adding an additional restriction on the initial displacement and the initial velocity,the invariant manifold with critical initial energy level is proved.Then all results obtained at the low initial energy level are extended to the critical initial energy level according to the proof process of the low initial energy level.In the case of supercritical initial energy,we derive the global existence of solution and finite time blowup by redefining new stable manifold and unstable manifold with their invariance.We study in detail a semilinear quasi-parabolic system with a non-local source(strong dissipation heat system).The heat system can be used to model phenomena in population dynamics and biological sciences,in which the total mass of a chemical or an organism is conserved.The nonlocal term acts to conserve the spatial integral of the unknown function as time evolves.Such equations give insight into biological and chemical problems where the conservation properties predominate.The subject first explains how non-local sources make the solution of the system remain constant in spatial integration,hence the comparison principle is no longer applicable here.This work overcomes this difficulty by using the energy estimation and concave function methods.Then we give the invariant manifold of the thermal system at low initial energy level by introducing the potential well structure.Based on this,the existence of global solution at low initial energy level is proved by the classical Galerkin method,and then the exponential decay of the norm in the phase space of global solution is proved by Gronwall’s inequality.Then,with the help of the invariant manifold and the concave function method,the finite time blowup of solution for the system at low initial energy level is proved.By scaling the initial values,all the results obtained at low initial energy level are parallelly extended to the critical initial energy level.At last,we prove that the solution of system blows up in finite time with supercritical initial energy level for the same initial conditions by two different methods.