Research On Well-posedness Of Solutions For Nonlinear Wave Systems Based On The Potential Well Theory

In the natural sciences,nonlinear dynamics has a very wide range of applications in many fields,providing a strong theoretical basis for reasonably solving the various relevant problems.The nonlinear wave equation is one of the most important nonlinear dynamical systems,which was a very important engineering application on the solution with respect to time,so it is a popular problem to many scholars.This paper aims to hold a deep study on the dynamic behavior of solutions for some nonlinear wave systems in the structure frame of potential well.This thesis undertakes a comprehensive consider on the global existence,asymptotic behavior and finite time blow up of the solution regarding the initial energy at three different energy levels(sub-critical energy level,critical energy level and arbitrarily positive initial energy level),and derives the influence of solution in arbitrarily initial data.This thesis undertakes a comprehensive study on the well-posedness of the global solution for the elastic beam system with nonlinear weak damping and linear strong damping.Firstly,we consider the existence and uniqueness of the local solution by using the Banach contraction mapping principle.Secondly,at the three different energy levels,it holds a comprehensive study on the behavior solution of the nonlinear wave system with complex dissipation term by employing the potential well theory.For the sub-critical and critical energy levels,it is analyzed in detail what kind of initial value affecting the solution of the system is global existence or blows up in the finite time.It is find that the dissipation makes a exponential decay of the energy for the system when we consider the long time behavior of the global solution by the energy method.For the arbitrarily positive initial energy level,the linear damping was studied instead of the nonlinear weak damping,and the existence or nonexistence of the global solution for the elastic beam system with the linear weak damping and strong damping was analyzed.Considering the fact that the beam is not only translational motion but also rotational motion during the vibration process,in order to explore the dynamical behavior of the solution for the non-planar rotating beam system(so the coupled elastic beam system),we studied the global well-posedness of the solution system at the three different initial energy levels,for the initial boundary value problem of coupled elastic beam system with complex dissipation term and coupled external force source term.The correlation analysis is carried out by utilizing thepotential well method,the Clerkin method,the boundedness theorem,the improved concave function method and the energy estimation.With the purpose of overcoming the difficulties of complex dissipative,dispersive,nonlinear and coupled terms to control the various functions in the potential well structure,by introducing the appropriate auxiliary function to make appropriate shrink estimate we have a discussion on the influence of initial value of different conditions on the solutions.This thesis undertakes a study on the global existence and finite time blow up of the solution for the nonlinear wave system with strong dissipative term,double dispersive term and initial value problem which describes the nonlinear longitudinal stress wave propagation inside the elastic cylindrical rod.We discuss the optimal conditions for the existence and non-existence of the global solution in the sub-critical,critical energy level,and we discuss the finite time blowing in the arbitrarily positive initial energy level.It is well known that the negative energy easily leads to the solution in finite time blowing.When the negative initial energy is extended to the positive initial energy,it is found that the system damping coefficient is narrowed.Then,we use the control function method to obtain the initial conditions for the existence of the global solution in the arbitrarily positive initial energy level and to promote the results of the existing sub-critical and critical energy levels.The class of high-order dispersive nonlinear wave systems with initial boundary value data can describe the transverse propagation of homogeneous waves on elastic-plastic rods under the action of nonlinear external source term.For the system with high-order and more general nonlinear source term,we have a deep consider on the global well-posedness of solution by using the potential well theory.It is found that by using the Calerkin method combined with the bounded principle in the sub-critical level,the existence of the global solution is proved when the initial value of the system falls into the potential well.By using the concave function method,the solution is blowing in finite time when the initial value of the system is outside the space of the potential well.Then,all the results in the sub-critical level are propagated parallel to the critical level.For the arbitrarily positive initial energy level,the existence of the global solution of the system is obtained by introducing the auxiliary controllable function and the differential control inequality.By using the concave function method,the nonexistence of the global solution is given when the initial value satisfies some kinds ofcondition.Based on the potential well theory,the dynamic behavior of the solution for the stress wave system with high order dispersion is analyzed under different energy states.