Existence And Stability Of Traveling Wave Fronts For Nonlocal Delayed Reaction-diffusion Equations

With the development of the society, it is widely used in chemical, engineering, ecolog-ical, financial etc. However, in the real world, the future state of the system depends on not only the present states but also the past states. Therefore, a more realistic model would be delayed reaction-diffusion equations which describes the development of the system. Consequently, the researches on delayed reaction-diffusion equations are very meaningful in the theory and applications.Based on the above reasons, in this thesis, we shall study the existence and stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations.Firstly, Chapter1introduces the research background and current situation of travel-ing wave fronts for nonlocal delayed reaction-diffusion equations. Then, this Chapter also presents contents and methods. Last, this Chapter gives some definitions and theorems.Changing the integral equation to the differential equation and using Schauder fixed point theorem in combination with upper and lower solution method, Chapter2shows the existence of traveling wave fronts φ(x+ct) for nonlocal delayed reaction-diffusion equations and the results in this chapter generalize the existing results.Choosing appropriate weighted functions, building two energy estimations and using the weighted energy method combined with the comparison principle, Chapter3shows the asymptotic stability of traveling wave frontsφ(x+ct) for the given equations. In addition, Chapter3also obtains the corresponding delayed rate. Immediately, where u(x, t) is the solution of the given equations, d>0.μ>0are some constants. In particular, in the sense of L∞norm, the solution u(x, t) converges to the traveling wave front φ(x+ct), immediately, Thus, the results in this Chapter are more valuable in the applications.