Qualitative Properties Of Traveling Fronts Of Reaction-Diffusion Equations

Reaction-diffusion equations play an important role in modeling the spatial-temporal patterns, and many researchers focus the properties of traveling wave solutions as they can describe finite spreading speed of propagation in nature, limited vibration phenomenon. As the form of traveling wave is simple and the solution itself is too special, the direct use of traveling wave solutions to describe the phenomenon of nature is very meaningful. Moreover, a detailed qualitative analysis of traveling wave solutions is very important. In the phase transition processes, the phenomena can be observed if the wave is stable in some sense.In this thesis, the qualitative properties of traveling waves for reaction-diffusion equations and their application in biological populations are studied, including uniqueness, stability and minimal wave speed.First of all, monotonicity and uniqueness of traveling waves of a class of biolog-ical models with stage structure are studied. Asymptotic behavior of the traveling waves as the variable tends to infinity is discussed, a formula about the decay rate is given. Furthermore, from comparison principle and sliding method, monotonicity and uniqueness of traveling waves are studied. As an application, for a class of epidemic models, the properties of traveling wave solutions are given.Secondly, in high-dimensional case, the properties of pulsating wave fronts in spatial and temporal periodic media are studied. For the KPP pulsating wave fronts in periodic media, uniqueness result provides a complete classification of KPP fronts. Global asymptotic stability is based on the construction of proper the sub-and super solutions which are very close to the front. It turns out that when time is large enough, the wave fronts is globally asymptotically stable in L∞space. It is worth noting that, a uniform estimates in the variables which are orthogonal to the direction of propagation is a tough process.Finally, monostable integral recursion in a periodic habitat is studied, especially for spreading speeds. A variational formula was given when the periodic tends to infinity. This result provides an important information in finding the minimal wave speed. For the bistable case, because the construction of proper upper and lower solutions is very tough, so global asymptotic stability is first derived by the squeezing technique based on comparison principle, and this result implies the uniqueness of the wavefront. Then the existence of bistable wavefront is established.