Asymptotic Stability Of Traveling Waves For Infinite-dimensional Lattice Differential Equations With Delay

With the progress and development of natural science, the application of the lattice differential equation is increasingly common. On the one hand, lattice differential equation is used to describe the model with discrete behavior in real life, such as the phenomenon of patches in population, crystal growth in material physics. On the other hand, it is necessary to make spatial discretization of partial differential equation in the practical application, namely the formation of lattice differential equations, such as in a computer data simulation. It is found that lattice differential equation has more complex dynamics behavior than continuous partial differential equation. Thus, it has important theoretical and realistic significance to study the lattice differential equation. In this thesis,we consider the stability of non-monotone traveling waves to a class of lattice differential equation with nonlocal diffusion and nonlocal delay.Firstly, we summarize the recent development of lattice differential equations and give a brief introduction to the problem of this thesis.Secondly, we prove the existence of non-monotone traveling waves when the birth function is non-monotone. Our methods are Schauder fixed point theorem and the upper limit principal.Lastly, we study the asymptotic stability of non-monotone traveling waves.We first introduce a suitable transform function to switch the equation to a new equation. Then we prove the local existence to the solution of this new equation by the standard iteration technique. Finally, we establish the asymptotic stability of the non-monotone traveling waves. The adopted approach is the transformed energy method combining with Fourier transform and nonlinear Halanay’s inequality.