Study On Properties Of Solutions For Two Classes Variable Coefficients Wave Equation With Balakrishnan-Taylor Damping

For the last decades,a great deal of mathematical effort has been devoted to the study of nonlinear partial differential equations,such as wave equations,heat equations,Schr¨odinger equations,due to their wide range of applications in physics,engineering,material science and other fields.The stability and blow-up of the solutions to the nonlinear partial differential equations become the important research topics.Especially in recent years,many authors were interested in wave equations with Balakrishnan-Taylor damping and variable coefficients.In this thesis,we mainly study the properties of solutions for two classes variable coefficients wave equation with Balakrishnan-Taylor damping.In section 1,we provide some research results about wave equation of variable coefficients with Balakrishnan-Taylor damping and source term,and variable coefficients viscoelastic wave equation with Balakrishnan-Taylor damping and acoustic boundary conditions.Furthermore we give some symbol description.In section 2,we discuss the asymptotic behavior of solution for the wave equation of variable coefficients with Balakrishnan-Taylor damping,time-varying delay and source term.By establishing some Lyapunov functionals which are equivalent to the energy of the system and using the potential well method to deal with the source term,we prove a general decay result of the energy of system.In section 3,we study blow-up of solution for the variable coefficients viscoelastic wave equation with Balakrishnan-Taylor damping,strong damping,source term and acoustic boundary conditions.By competing nonlinear source term with viscoelastic term,Balakrishnan-Taylor damping term and strong damping term,we prove that under suitable conditions on the initial data,the relaxation function,the elements of the variable coefficients matrix and the order of source term,the solution of system blows up in finite time.