Traveling Wavefronts Of Delayed Differential Equations With Global Interaction

This thesis is concerned with the existence, uniqueness and stability of the traveling wavefronts of the delayed differential equations with global interaction.In Chapter 1, we first recall several problems related to dynamical systems, which is important in theory and application of dynamical system. The main issues are: reaction-diffusion equations,.lattice dynamical systems, traveling wavefronts and delayed dynamical systems. Motivated by these, we will study the traveling wavefronts of delayed differential equations with global interaction in this dissertation.In Chapter 2,we study the traveling wavefronts of nonlocal diffusion system with delays. First, we establish an integral operator to transform the existence of traveling wavefront into the existence of fixed point of the operator. According to the operator and the upper-lower solutions, we give a monotone sequence by monotone iteration if the nonlinearity satisfies the quasimonotone and exponentially quasimonotone conditions. By passing the sequence to a limit, we get the existence of traveling wavefront. To illustrate our theory, we consider the nonlocal diffusion Logistic equation and Belousov-Zhabotinskii system, which implies our theory is applicable and the conditions are easy to be satisfied.In Chapter 3, we are concerned with the existence of the traveling wavefronts in a global interaction lattice dynamical system with time delays. We first establish an equivalent problem by a proper integral operator. By discussing the fixed point of the equivalent problem, we get the existence of traveling wavefronts of the lattice dynamical system. More precisely, we study the fixed point of the integral operator by upper-lower solutions and Schauder's fixed point theorem when the system satisfies quasimonotone or exponentially quasimonotone conditions, which reduces the existence of traveling wavefront to the existence of a suitable pair of upper-lower solutions that are easy to be constructed in practice. Finally, we give an example to illustrate our theory, which implies our conditions are easy to be satisfied.In Chapter 4, we investigate the existence and uniqueness of the traveling wavefronts of a class of nonlocal diffusion equations with delays. By the main results in Chapter 2, we give out the existence of the traveling wavefronts and the asymptotic approximation of a kind of general differential equations. We give out the compari-son principle by studying the corresponding initial value problem and utilizing the theory of operator semigroup. Using the comparison principle, upper-lower solution and the squeezing technique, we prove the stability of the traveling wavefronts of the equations.