Time-Periodic Solutions And Traveling Wave Solutions To The Three-Dimensional Nonlinear Viscoelastic Dynamic System

The study on viscoelastic dynamic system is an important subject in both theory and applications.From the end of 1950s to the early of 1960s,Coleman and Noll et al developed the constitutive theory of viscoelastic materials with memory. Especially,the article "Thermodynamic of materials with memory"[3],given by Coleman in 1964,exerted a great influence on this domain.This thesis deals with viscoelastic materials,with memory,of which the constitutive equation is a single-integral law.The main aim of the thesis is to prove the existence of time-periodic solutions and nontrivial traveling wave solutions to the three-dimensional viscoelastic dynamic system.At present,there have been some important works on the existence of global solutions to the nonlinear viscoelastic dynamic system,but the works on the existence of periodic solutions and traveling wave solutions are lacking.In 1991,for viscoelastic solid materials,using viscosity regularization and compensated compactness method,Feireisl E.[8]proved the existence of periodic weak solutions to the one-dimensional nonlinear viscoelastic dynamic equation in a special case.In 1992,by studying a general linear integro-differential equation,Qin T.H. [26]proved the existence of periodic solutions to the one-dimensional linear viscoelastic dynamic equation in the case of viscoelastic solid materials.In 1997,Qin T.H.[28]proved the existence of periodic solutions to the one-dimensional semilinear viscoelastic dynamic system.Moreover,by Galerkin method,and employing the singularity of integral kernel,Qin T.H.[29]proved the existence of periodic weak solutions to the one-dimensional nonlinear viscoelastic dynamic system for viscoelastic solid and liquid materials,respectively.All these results dealt only with the onedimensional viscoelastic equation,we extend them to a general three-dimensional nonlinear viscoelastic system. In 1976,for the viscoelastic materials exhibiting long range memory,by monotonic methods,Greenberg J.M.[9]proved the existence of traveling wave solutions to the one-dimensional nonlinear viscoelastic dynamic equation.In 1988,for the viscoelastic materials with fading memory,Liu T.P.[21]proved the existence of smooth and nonsmooth traveling wave solutions to the one-dimensional nonlinear viscoelastic dynamic equation.In 2003,for the viscoelastic materials with a special integral kernel,by virtue of a higher-order iterative process,Qin T.H.and Ni G.X.[31]proved the existence of traveling wave solutions to the three-dimensional nonlinear viscoelastic dynamic system.We extend this result to the case where the constitutive equation is a general single-integral law.The arrangement of this thesis is as follows:First of all in Chapter 1,we give the development of viscoelasticity and main results in this paper.In Chapter 2,we discuss time periodic solutions to a general three-dimensional nonlinear viscoelastic system with Dirichlet boundary condition.By Galerkin method, and employing the singularity of integral kernel,we obtain the energy estimates in Sobolev spaces with fractional index.Moreover,by employing interpolation theorems and compactness theorems in Sobolev spaces,we show the existence of the solutions to the problem for viscoelastic solid and liquid materials,respectively.In Chapter 3,for the viscoelastic materials with the constitutive relation in a general single-integral law,under certain hypotheses,if the speed of propagation is between the speeds determined by equilibrium and instantaneous elastic tensors,in virtue of a higher-order iterative process and the Schauder's fixed point theorem, we get the existence of nontrivial traveling wave solutions to the three-dimensional nonlinear viscoelastic dynamic system.