Asymptotic Stability Of Traveling Waves With Critical Speeds For Reaction-convection-diffusion Equations

This thesis is mainly concerned with the asymptotic stability of traveling waves with criti-cal speeds for reaction-convection-diffusion equations by using pointwise estimates.It is divided into five chapters.The background and recent advances on traveling waves with critical speeds for reaction-convection-diffusion equations are introduced in Chapter 1,the main results of the asymptotic stability of traveling waves with critical speeds for reaction-convection-diffusion equations are presented.In Chapter 2,we mainly study the essential spec-trum of the linear operator,obtain the asymptotic estimation of the solutions of the eigenvalue problem of the linear operator and proved that the linear operator has no eigenvalue in the right half complex plane,finally we present the properties of Evans function.In Chapter 3,estimates of Green's function and its derivative are got.In the case of near?=0,the asymptotic estimate of the solutions of the eigenvalue problem of the linear operator are applied to get the estimate of Green's function and its derivative.In Chapter 4,we estimated the parabolic Green's function and its derivative,through the selection of the appropriate contour,the parabolic Green's func-tion and its derivatives decay about t at an algebraic rate are proved.In Chapter 5,it is proved that when the time t tends to infinity,the perturbations decays by t^-32 and the nonlinear stability of the traveling wave solution was obtained.Finally,the thesis was summarized,and proposed the future research directions.At present,there are not many research results on the asymptotic stability of the traveling wave of the critical speeds of the reaction-convection-diffusion equation with slow exponen-tial decay.The asymptotic stability of the traveling wave of the critical speeds of the reaction-convection-diffusion equation that we obtained in this paper generalizes the related research results of the traveling wave of the critical speeds of the reaction diffusion equation in the relat-ed literature.