Qualitative Study Of Solutions For Some Nonlinear Evolution Equations

Due to the wide application of the dynamic behavior of the solutions of nonlinear evolution equations in realistic nonlinear models,the existence and properties of the solutions have become a subject of great concern to mathematicians and physicists.In this thesis,we mainly study the existence,stability and blow-up of the solutions for several classes of nonlinear evolution equations.This thesis is divided into four chapters:In first chapter,we mainly introduce the background,the concepts of orbital stability,asymptotic stability and the research status.In addition,the definitions and theorems related to this thesis,as well as the main results,are also given.In second chapter,we consider the existence and orbital stability of the solitary wave solutions for the coupled Schrodinger-KdV system.When the parameters satisfy the suitable conditions,by applying the classical theory developed by Benjamin^[1]and Bona^[2]and the spectral analysis method,we obtain the existence and orbital stability of the solitary wave solutions.The results obtained in this thesis can be regarded as a supplementary extension of the results by Albert,Angulo^[3,4]and Chen^[5].In third chapter,we investigate the blow-up and asymptotic stability of the solution for a class of variable coefficient wave equation with nonlinear damping and logarithmic source.Firstly,the existence and uniqueness of local weak solution can be obtained by using the Galerkin method and contraction mapping principle.However,the long time behavior of the solution is usually complicated and it depends on the balance mechanism between the damping and source terms.When the damping exponent（p+1）（see assumption（H3））is greater than the source term exponent（q-1）（see the above equation）,namely,p+2>q,we obtain the global existence and accurate decay rates of the energy for the weak solutions with any initial data.Moreover,whether the weak solution exists globally or blows up in finite time,it is closely related to the initial data.In the framework of modified potential well theory,we construct the stable and unstable sets（see（3.10））for the initial data.For the initial data belonging to the stable set,we prove that the weak solution exists globally and has similar decay rates as the previous results.For p+2<q and the initial data belonging to the unstable set,we prove that the weak solution blows up in finite time for a little special damping g（u_t）=|u_t|^pu_t.In fourth chapter,we consider a damped beam equation with logarithmic source and viscoelastic term.By using Galerkin method,logarithmic Sobolev inequality and some compactness arguments,the local existence of weak solutions is obtained.Furthermore,we show the global existence and decay estimate of related energy under the smallness assumption on the initial data.