Traveling Wave Fronts And Entire Solutions Of Reaction-Diffusion Equations With Nonlocal Delays

Since the 1970s, there have been intensive developments in the theory of traveling wave solutions of parabolic differential equations. It was found that traveling waves can well model the oscillatory phenomenon and the propagation with finite speed of nature, so the existence, uniqueness and stability of traveling wave solutions have been widely studied. Due to time delays which usually exist in nature, there have been a number of works devoted to the study of partial functional differential equations (especially, delayed reaction-diffusion equations) from the dynamical systems and semigroups point of view since the 1970s. However, it has become recognized that the delayed reaction-diffusion equations can not accurately describe the spatial-temporal patterns of the objects of study. For example, consider a biological population. Since individuals are moving around, they have not been at the same point in space at previous times, which results in the spatial nonlocality. Since then, many delayed diffusion systems with nonlocal effects (reaction-diffusion equations, lattice differential equations) have been proposed to bring the models closer to the biological reality in population biology, spatial ecology and disease spread. Since such systems generally have the nonlinearities involving a weighted spatial averaging over the whole of the infinite spatial domain and the whole of the previous times, we call them reaction-diffusion systems with nonlocal delays. Though the systems are closer to the reality, the time delay and spatial nonlocality lead to many mathematical difficulties and essential changes of dynamics. For example, comparison theorems are not, in general, applicable for reaction-diffusion equations with nonlocal delays and the backward continuation of solution semiflows is very difficult. The time delay can lead to the change of stability of equilibrium and result in oscillations and chaos. The time delay can also slow the minimal wave speed of traveling wave fronts, make the traveling wave fronts lose the monotonicity and cause oscillations and periodic traveling wave solutions. In particular, the solution semiflows are not usually compact when the nonlocal term is incorporating into the equations. Therefore, it is not only more meaningful and valuable in theory and practice, but also more challengeable in mathematics to study such equations. This thesis is continuous to study such equations from the dynamical systems point of view, attempt to develop some new approaches for such equation and establish some new abstract results. For some specific models, investigate the influences of the time delay and spatial nonlocality, in particular, spatial nonlocality, on the dynamics of the equations and find some properties which were not reported previously. This thesis is mainly concerned with traveling wave solutions and entire solutions in reaction-diffusion equations (systems) with nonlocal delays. We firstly study the existence of traveling wave fronts of reaction-diffusion systems with nonlocal delays. By introducing different monotonicity conditions for the nonlinear-ity, and the concepts of G-compactness and M-continutity, we establish the existence of solutions for a class of abstract second-order mixed functional differential systems. Our methods are to use monotone iterations together with super- and subsolutions techniques and a non-standard ordering to develop a new monotone iteration scheme. We then apply the results to the corresponding wave systems and obtain the existence of traveling wave fronts. As a application for the main results, we carefully study the existence of traveling wave fronts for a single-species diffusive model with nonlocal delay and obtain some existence criteria of traveling wave fronts by choosing different kernels.We further study the existence, uniqueness and asymptotic stability of traveling wave fronts of the quasi-monotone reaction advection diffusion equations with nonlocal delay. We consider two case for the nonlinearity, that is, the monostable nonlinearity and the bistable nonlinearity. For the monostable case, the existence of traveling wave fronts is obtained by using the results established for the existence of traveling wave fronts of reaction-diffusion systems with nonlocal delays and the asymptotic stability of traveling wave fronts with phase shift is proved via employing squeezing technique together with the comparison principle. Furthermore, we show a priori asymptotic behavior of traveling wave fronts in minus infinity and obtain the non-existence of traveling wave fronts at the same time, then the uniqueness of traveling wave fronts (up to translation) follows from the asymptotic stability. In particular, we find that the time delay can slow the spreading speed and the spatial nonlocality can increase the spreading speed. For the bistable case, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique traveling wave front (up to translation, the wave speed is also unique), which is monotonically increasing and globally asymptotically stable (with phase shift). The influence of advection on the propagation speed is also considered for both two cases. When applied to some population and epidemiological models, we obtained many meaningful results in practice.Finally, using the increasing travelling wave solution estalished for the quasi-monotone reaction advection diffusion equations with nonlocal delay and the comparison argument, we prove the existence of entire solutions for the quasi-monotone reaction-diffusion equations with nonlocal delay. A key idea for the proof of the existence of entire solutions is to characterize the asymptotic behavior of the solutions as t→-∞in term of appropriate subsolutions and supersolutions. For the bistable equation, we further show that such an entire solution is unique (up to space-time translations) and Lyapunov stable. For the monostable equation, by considering the mixing of traveling wave solutions and the solution independent of spatial variable, other new types of entire solutions are obtained. Moreover, some sufficient conditions are given to ensure the existence of the solutions independent of the spatial variable via using the theory of monotone dynamical systems. As applications of the main results, we investigate entire solutions two reaction-diffusion equations with nonlocal delay for the monostable nonlinearity and bistable nonlinearity, respectively.