Qualitative Research On Some Classes Of Nonlinear Boussinesq Systems

The research topics of the present thesis are from the research projects supported by the National Natural Science Foundation of China(11101102,11471087).This thesis aims to undertake a deep study on the structure of potential well and tries to make a breakthrough on the potential well theory,which will be used to treat the nonlinear Boussinesq systems of high order and some nonlinear wave systems with special terms.This thesis considers the conditions on the global existence,long time behaviour and finite time blow up of solutions for above systems and the relationships between these conditions.What's more,this thesis undertakes a comprehensive study on the dynamic behavior of solutions regarding the initial energy at three different energy levels(sub-critical energy level,critical energy level and sup-critical energy level).Moreover this thesis will reveal the relationships between the corresponding coefficients of the structure of potential well and the conditions on the initial data,and also the influence on the nature of solutions.This thesis undertakes a comprehensive study on the global well-posedness of solutions for a class of six-order dispersive Boussinesq systems with combined nonlinear source terms at three different energy levels.The origin model of this Boussinesq system was introduced to describe the bi-directional propagation problem for small amplitude of shallow water wave and the dynamic behavior of nonlinear lattice in the elastic crystal.By using the Galerkin method and concavity method,the existence and non-existence of global solutions at sub-critical initial energy level are presented in the frame work of potential well.With the aid of scale transform we generalize the conditions on initial data assuring the existence of global solutions from the sub-critical initial energy case to the critical initial energy case.Moreover by introducing a subspace of potential well,we present some sufficient conditions on initial data such that the weak solution exists globally at sup-critical initial energy level,with the help of the technique of anti-dissipativity and the relationship between controllable functionals as well as the regularity of solutions together with the boundedness of solutions.Moreover the global non-existence of solutions for this problem at sup-critical initial energy level are proved by introducing a new auxiliary controllable function together with concavity method.This thesis is concerned with the global existence and non-existence of global solutions for certain nonlinear Boussinesq systems of sixth order with nonlinear generalized source terms at three different initial energy levels as the initial energy varies at the whole real space,which were proposed to analyze and model the motion of water wave with surface tension and describe the bi-directional propagation problem of shallow water wave.By using the Galerkin method and boundedness theorem,we show that solutions with sub-critical initial energy exist globally when the initial data belong to the stable set.And solutions with sub-critical initial energy can not exist globally together with the concavity method and the analysis of the relationship between the depth of potential well and the H~1norm of solutions when the initial data belong to the unstable set.Besides we show that the results about the global well-posedness of solutions at the critical initial energy level can be obtained similar to the sub-critical case.Furthermore this thesis makes a first try to handle the existence and non-existence of global solutions for certain Boussinesq systems with nonlinear generalized source terms at sup-critical initial energy level.This thesis puts forward a class of generalized dissipative Boussinesq systems with linear pseudo-differential operators to integrate the structures and results of numerous known non-linear dissipative Boussinesq systems,and makes a first tempt to consider the well-posedness of the local solution and the existence and non-existence of global solutions for this problem at different initial energy levels.By employing the Cauchy-Schwarz inequality to control the nonlinear source terms and applying the contraction mapping principle,we obtain the exis-tence and uniqueness of local solution.In the frame of potential well we present that the initial displacement locating the potential well ensures the existence of global solutions by utilizing compactness method and boundedness theorem and scaling the norm of operators together with controllable functions.On the contrary,the initial displacement locating the outer space of po-tential well forces the non-existence of global solutions by using the relationship between the depth of potential well and the norm of operator as well as the concavity method.Moreover the existence of global solutions with critical initial energy are obtained by constructing a series of approximate solutions.And by using the auxiliary function introduced in the sub-critical initial energy case we prove the non-existence of global solutions at the critical initial energy level.Furthermore we derive a sufficient condition on the initial data with sup-critical initial energy such that global solutions of this problem do not exist globally,which generalizes the results on the non-existence of global solutions with sub-critical and critical initial energy for dissipative Boussinesq systems obtained in related works.This thesis investigates the existence and non-existence of global solutions as well as the long time behavior of global solutions for a class of fourth-order strain wave systems with nonlinear damping term.The original model of this system can describe the longitudinal dis-placement of viscous elastic-plastic rod and viscosity plastic microstructure of the plane shear.By constructing the structure of potential well and utilizing the variational method as well as techniques of estimations on functionals and control theory for such nonlinear strain wave sys-tem with dissipative terms and high-order term,we obtain the sharp condition for the existence and non-existence of global solutions at sub-critical initial energy level.For the critical ini-tial energy case we undertake comprehensive discussions on the existence and non-existence of global solutions,and present some sufficient conditions on initial data to ensure the global non-existence of solutions without the positive definite initial inner product.For the sup-critical initial energy case,by employing the technique of anti-dissipativity and auxiliary controllable functions,we obtain the existence of global solutions together with the boundedness principle and the regularity of solutions,and the non-existence of global solutions by means of concavity method.By exploiting the multiplier method on energy functions and controlling accurately the norm of the structure terms of this system,we make a first try to find out that the decay rate of global solutions for certain nonlinear strain wave systems are dependent on the power of the strain term and the dissipative term.This thesis undertakes a deep discussion on the existence,long-time behavior and non-existence of global solutions for a class of nonlinear fourth-order dispersive wave systems with strong dissipative term and nonlinear weak damping term at the whole initial energy level.By constructing the variational structure of system and utilizing the Galerkin method and bound-edness theorem,we prove that global solutions exist at sub-critical initial energy level.And the exponential decay of global solutions for this system is present under the condition that the initial data lie in the potential well and the initial energy is controlled by the depth of potential well by introducing the perturbational controllable energy function and control inequalities as well as the accurate computation of the depth of potential well.We obtain the non-existence of global solutions with the sub-critical initial energy under the condition that the initial energy is controlled by the depth of potential well by employing the concavity method and the rela-tionship among the potential energy functional and the total energy functional together with the relationship between the depth of potential well and the H~2norm of solutions starting from the expression of the depth of potential well.And the conclusions obtained in the sub-critical energy case are extended to the critical energy case by using the boundedness principle and the idea of scale-change as well as the concavity method.In addition,by means of differential-integrable controllable inequalities and the relationships among the controllable functionals yielded by systems we prove that certain solutions exist globally at sup-critical initial energy level.Further by employing the concavity method as well as the technique of anti-dissipativity,we show that for which initial data global solutions of this system do not exist at sup-critical initial energy level.