Traveling Wave Fronts And Their Properties Of Some Time Delayed Reaction Difusion Equations

Since the1970s, there have been intensive developments in the theory of travelingwave solutions of parabolic diferential equations. It was found that traveling waves canwell model the oscillatory phenomenon and the propagation with finite speed of nature,so the existence, uniqueness and stability of traveling wave solutions have been widelystudied. Due to time delays which usually exist in nature, there have been a number ofworks devoted to the study of partial functional diferential equations (especially, delayedreaction-difusion equations) from the dynamical systems and semigroups point of viewsince the1970s. This thesis is mainly concerned with traveling wave fronts in delayedreaction-difusion equations. The main work of this dissertation are as follows.In Chapter1, we mainly introduce the background, research developments, andachievements of our research objects. In the end, we give a simply introduction of themain content of this paper.In Chapter2, we study traveling wave fronts for a degenerate difusion equation withtime delay. We first establish the necessary and sufcient conditions to the existenceof monotone increasing and decreasing traveling wave fronts, respectively. Moreover,special attention is paid to the asymptotic behavior of traveling wave fronts connectingtwo uniform steady states.In Chapter3, we consider a modified disease model with distributed delay. Theexistence of traveling wave fronts connecting the zero equilibrium and the positive equi-librium is established by using an iterative technique and a nonstandard ordering for theset of profiles of the corresponding wave system. We also study the critical wave speedand give a detailed analysis on its location and asymptotic behavior with respect to thetime delay.In Chapter4, we consider an n-dimensional delayed system with nonlocal difusionand mixed quasimonotonicity. By developing a new definition of upper-lower solutionsand a new cross iteration scheme, we establish some existence results of traveling wavesolutions. These results are applied to a nonlocal difusion model which takes the four-species Lotka-Volterra model as its special case.