Some Qualitative Problems Of Nonlinear Diffusion Equations With Delay(s)

In this monograph, we study some qualitative problems of non-linear diffusion equations with delay(s), which come from a variety of delay diffusion phenomena appearing widely in nature, such as spread of infectious disease, the laser model in physicsand and so on. S-ince nonlinear diffusion models with delay(s) have extensive practical background and important theoretical value, in recent years, the in-vestigation in this direction are attracting more and more attention of mathematicians, and become one of the hot topics both inside and outside the country. The aim of this paper is to study the periodic problem of the model, the long time asymptotic behavior of solutions and travelling wavefronts respectively.Chapter 1 is the exordium of this paper.In Chapter 2, we firstly consider the periodic problem of Nichol-son's blowflies model with non-Newtonian polytropic filtration diffu-sion subject to homogeneous Dirichlet boundary value condition. Up to now, we have not found any paper about periodic problem of Nichol-son's blowflies model with nonlinear diffusion. By constructing some suitable Lyapunov functionals, the a priori estimates on all possible periodic solutions, and combining with Leray-Schauder fixed point theorem, we finally establish the existence of time periodic solution-s. Secondly, for periodic problem of degenerative parabolic equations with multiple delays, we establish the existence of periodic solutions.In Chapter 3, we firstly investigate the asymptotic stability of the positive steady state solutions for Nicholson's blowflies model with p-Laplacian under homogeneous Dirichlet boundary value condition and nonnegative initial value condition. Recently, many researcher-s discussed the asymptotic stability of the steady state solutions for Nicholson's blowflies model with linear diffusion. Our aim is to study the case with nonlinear diffusion. Using the method of upper and lower solutions and its associated monotone iterations, we establish the exis-tence, uniqueness and the asymptotic stability of positive steady state solutions. Secondly, we study the attractability of the periodic solu-tions for Nicholson's blowflies model with non-Newtonian poly tropic filtration diffusion subject to homogeneous Dirichlet boundary value condition, and obtain the existence of the attractor for this model.In Chapter 4, we study the traveling wavefronts of Nicholson's blowflies model with nonlocal delay. In recent years, the works of the former researchers concentrated on the traveling wavefronts for one-demensional model with simple kernel functions. We are inter-ested in establishing the existence of the multi-dimensional traveling wavefronts with three different complicated kernel functions, and in-vestigate the convergent rate of travelling wave solutions at infinity.The conclusion are summarized in the last chapter.