Stability Analysis Of Two Kinds Of Evolutionary Equations

Partial differential equations,an important branch of modern mathematics from all kinds of real life,are widely used in the natural science,physics and engineering fields,and so have great significance and value.A large number of researchers have devoted much attention to these equations for a long time.Wave equation and heat equation,the most basic and important kinds of partial differential equations,have always been the research problems of great interest to the authors.This thesis mainly studies the effect of Kelvin-Voigt damping on spectrum analysis of a wave equation and the stability of a heat equation in non-cylindricalr domains,many important results have been obtained in the research of these two kinds of problems.This thesis makes a more systematic and in-depth study and analysis of these two parts by using the conclusions already obtained,and draws some new conclusions.The thesis consists of three sections.In Chapter 1,we provide a simple research summary of the stability of wave equation and heat equation and main research problems of this chapter.In Chapter 2,we consider the following the wave equation with a small amount of Kelvin-Voigt damping(?)where c>0 is a system parameter and d>0 is a small Kelvin-Voigt damping coefficient.We give a detailed spectrum analysis of the system operator,from which we show that the generalized eigenfunction forms a Riesz basis for the state Hilbert space.That is,the precise and explicit expression of the eigenvalues is deduced and the spectrum-determined growth condition is established.Hence the exponential stability of the system is obtained.In Chapter 3,We consider the following the heat wave equation in non-cylindrical domain(?)write (?)_T~K for the non-cylindrical domain {(x,t)E R2;0<x<?k(t),t?(0,T)},w(x,0)=w0(x)is initial condition.the chapter applies the Lyapunov function to calculate the energy derivative that does not contain boundary conditions,and then leads to the boundary conditions.By using Poincaré inequality,we obtain the condition of the heat equation stability in non-cylindrical domain,and then we get the exponential stability of the heat equation.