The Research On Well-posedness For Two Classes Of (Coupled) Nonlinear Wave Equations

This thesis mainly studies the well-posedness of solutions at three different initial energy levels(subcritical initial energy level,critical initial energy level and supercritical initial energy level)for two classes of(coupled)wave equations with nonlinear damping and nonlinear source terms,and aims to reveal the relationship between the choice of initial data and the well-posedness of solutions and further to develop the potential well theory.Chapter 2 undertakes a comprehensive study on the well-posedness of solution for a class of wave equation with nonlinear strong and weak damping and nonlinear source term.The physical model of this problem can be described the longitudinal motion of a viscoelastic structure,and a system about the longitudinal movement of a viscoelastic configuration that conforms to the nonlinear Voight model.Firstly,this chapter introduces the total energy functional,potential energy functional,Nehari functional,in-well(stability)sets and out-of-well(instability)sets to construct the corresponding framework of potential well theory and obtain the depth of potential well as well as some relative lemmas.Secondly,at the subcritical initial energy level,the existence of global solution is given by using the Galerkin method and the finite time blowup of solution is obtained with the improved concave function method.Furthermore,the asymptotic behavior of the global solution is also discussed with the help of Gronwall inequality and interpolation inequality.Thirdly,the conclusions at the subcritical initial energy level are extended to the critical initial energy level with the idea of scaling.Finally,this chapter seeks some initial data and constructs appropriate auxiliary functional to obtain the finite time blowup of solution at the supercritical initial energy level.At last,the lower bound of the blowup time at arbitrarily positive initial energy level is also estimated.Chapter 3 mainly studies the initial boundary value problem of a class of coupled wave equations with nonlinear strong and weak damping and nonlinear source term.This chapter focuses on how the coupling phenomenon of two wave equations with nonlinear strong and weak damping and nonlinear source term affects the well-posedness of the solution.At first,the existence of local solution is obtained by using the Galerkin method combined with the principle of compressed mapping.And then,this chapter introduces the total energy functional,potential energy functional and Naheri functional to construct the framework of potential well theory.At subcritical and critical initial energy level,the global existence and the finite time blowup of the solution are obtained by using Galerkin method and improved concave function method,respectively.This chapter also discusses the asymptotic behavior of the global solution at the subcritical initial energy level by using important inequalities such as Gronwall inequality.At last,the finite time blowup of solution at arbitrarily positive initial energy level is given by introducing new auxiliary functional combined with the improved concave function method.