Existence Of Traveling Wavefronts For Reaction Diffusion Equation With Distributed Delay

The theory of travelling wave solutions of parabolic differential equation is one of the fastest developing areas of modern mathematics and has attracted much attention due to its significant nature in biology, chemistry, epidemiology and physics. Travelling wave solutions are solutions of special type and can be usually characterized as solutions invariant with respect to transition in space. From the physical point of view, travelling waves describe transition processes. These transition processes(from one equilibrium to another) usually don't consider their initial conditions and the properties of the medium itself. Among the basic questions in the theory of traveling waves, the existence of travelling wave solutions is an important objective.As deeply understanding the nature, people find that introducing delay effect in some classical and modern mathematical models better accords with the fact. At the same time, people's knowledge of delay also realizes transition from the discrete delay to the distributed delay and accuracy from the local distributed delay to the nonlocal distributed delay.This article mainly discusses the existence of travelling wavefronts for the generalized Burgers-Fisher equation with distributed delay. By using the geometric singular perturbation theory, we gain the existence of travelling wavefronts with local and nonlocal distributed delay, deepening the understanding of Burgers-Fisher equation.