Existence And Stability Of Reaction-Diffusion Equation With Delay

In recent years, Partial differential equation has been extensively and intensively used in innovative and high technology, economy, management and various of engineering problems, and all these specific applications are achieved by establishing mathematical models, meanwhile, most mathematical models come down to reaction-diffusion equations, as a result, great importance is increasingly attached to reaction-diffusion equation. Traveling wave solution not only can perfectly describes the phenomenon of biological invasion, disease transmission and so on, but also reveals many important characters of reaction-diffusion equations. This thesis mainly deals with the existence of traveling wave solutions of partially degenerated reaction-diffusion equations with delay and the exponential stability of traveling wave solutions for a 2D lattice delayed differential equation with global interaction.Firstly, we study partially degenerated reaction-diffusion equations with delay, by establishing a proper operator and using Schauder’s fixed theorem, the existence of the traveling wave is reduced to that of an admissible pair of upper and lower solutions.Then by constructing an appropriate pair of upper and lower solution, the abstract results are applied to a class of reaction-diffusion population model with delay and quiescent stage, and the existence of traveling wave fronts is obtained.Secondly, we study a two-dimensional(2D) lattice delayed differential equation with global interaction. By constructing a proper weight function and establishing weighted-estimated, we prove that any given traveling wave front with large speed is global time-asymptotically exponential stability by applying the comparison principle combined with the technical weighted-energy method.