Studies On The Asymptotic Behavior And Steady States Of Two Kinds Of Nonlinear Parabolic Equations

In this dissertation,we consider the asymptotic behavior and steady states of two kinds of nonlinear parabolic equations.Firstly,we consider the global existence and blow-up conditions of the porous medium equation,which were studied extensively in previous years when the initial energy E(u0)<d,where E(u0)represents the initial energy and d is a positive constant which will be given in the main part of the dissertation.We complete the previous studies by studying the case E(u0)= d.Secondly,we consider a Lotka-Volterra predator-prey model with cross-diffusion of fraction type.By analyzing the eigenvalue problem of the linearization problem,bifurcation theory and topological degree theory,we get the properties of the positive steady-state solutions,and the multiplication of the positive steady-state solutions.The results generalize and complete the previous results.Furthermore,we show the influence of the cross-diffusion coefficients on the coexisting region and the local bifurcation of the limit system.The dissertation is divided into three parts:· In the first chapter,we introduce the background the porous media equation and the Lotka-Volterra predator-prey model with cross-diffusion of fraction type,and the innovation of this dissertation.· In the second chapter,we study the global existence and blow-up conditions of the porous medium equation when the initial energy E(u0)= d.· In the third chapter,we study the multiplication of the positive steady-state solutions and influence of the cross-diffusion coefficients on the coexisting region to the Lotka-Volterra predator-prey model with cross-diffusion of fraction type.