Qualitative Problem Of Nonlinear Reaction Diffusion Equations With Delay

Nonlinear reaction-diffusion equations, as an important class of parabolic equations, come from a variety of diffusion phyenomena appeared widely in nature. In diffusion phenomena, the variety of some things depend on not only current but also the past states, we call this time lags as delay. In 1996, Volterra applied a delayed model when he studied the dynamics of biological groups. Then people are suggested as delayed models in many cases such as the infection spread phenomenon, the order car driving and the laser model in physics. The foundation and development of delayed models enrich the theory of biochemistry, dynamics of biological gronps, physics and so on. Presently, there have been many valuable results on linear and nonlinear diffusion equations with delays. All of these are applied widely in the field of ecology, physics and random problems. Therefore, the study on nonlinear reaction-diffusion equations with delay possess very important theoretical value and practical sense.The delayed heat equation with nonlocal sources is an important kind of nonlinear reaction-diffusion equations, which are the equations with delay and spacial nonlocal effects. In ecology, this model mainly describe the effects produced by the individuals around during the group moving. There have been a tremendous amount of papers on delayed heat equations with nonlocal sources. Most of scientific research results are mainly about the corresponding properties of travelling wave fronts and steady states. The theory of travelling wave fronts of parabolic equations is one of the fast developing fields in modern mathematics, which attracts a large number of mathematicians both in China and abroad, because of whose importance in biology, chemistry, epidemiology and physics. In physics, travelling wave fronts describe transfering process from a equilibrium state to another. In the infectious disease model, travelling wavefrots denote the infection spread with a constant wave spead. There have been a tremendous amount of papers about the travelling wave fronts of delayed heat equations with nonlocal sources. As early as 1990, Britton introduced the nonlocal delayed model:In his paper, Britton obtained the existence of the travelling wave-fronts of this equation. Later, the researchers studied the special and expending forms extensively, and got many important results about travelling wavefronts (see[2][9][13] etc). For other forms of delayed heat equations with nonlocal sources, many scholars applied different methods to explore those equations and gained many valuable investigations. Besides travelling wavefronts, the steady states are also the important object which the researchers study for. The steady states usually describe the case that something reach a steady state. Because of their profound physical background, they are the favour of researchers. In 1996, in [26], S.A.Gourley proved the stability of steady states of the system:The author extended the corresponding result of the model in [1]. Horst. R, Khalid Boushaba and other researchers discussed properties of steady states of other delayed heat equations with nonlocal sources separately.The heat equations with spatio-temporal delays are the delayed heat equations with nonlocal sources with especial forms, which possess especial properties. They are heat equations which are nonlocal in both spatio and temporal. Usually, the equations with spatio-temporal delays are used to describe evolutions of groups. The spatio-temporal delayed models attract researchers' attention gradually and become one of the hotspots in the field of partial differential equations. The earliest and the most representative equation with spatio-temporal delay is the model:which was proposed by Britton in [1], in 1990. Britton studied the existence of travelling wavefronts in that literature. Subsequently, many researchers considered the similar models. Zhicheng Wang, Shangbing Ai etc studied the properties of travelling wave fronts of equations or systems with spatio-temporal delay with other forms separately. The steady states of heat equations with spatio-temporal delay also have the similar properties such as existence, stability, which can be found in [18]. According to the continuity of delays, delayed equations could be divided into equations with discrete delays and distributed delays. In the rich and colorful natural phenomena in the boundless universe, the phenomena described by these two kinds of delayed equations exist extensively. The researchers pay attention at all times to the heat equations with discrete delays, which are applied in many fields. In early literatures, both Jack K.H in 1986 in [37] and Shiwang Ma in 2001 in [38], they discussed the delayed equation model:and got the asymptotic behavior of solutions, existence of travelling wavefronts and so on. In addition, J.Huang, Yifu Wang, Guosheng Zhang and the other people studied the equations with discrete delays and obatained many important and profound results. The heat equations with distribute delays are the equations in which the delayed terms are nonlocal, which can describe the average value in temporal. There are many distribute delays phenomena in nature. Many mathematicians explore various the heat equations with distribute delays and obtained significant theoretical and practical results, see [39] [40] [43] etc.Besides the heat equations, many researchers have studied delayed diffusion equations with the general form. The so-called general form is that the diffused term is similar to the following form:A lot of practical problems, in biology, ecology and so on, could be discribed by these equations. These models are applied to signal processing, image processing, pattern classification, quadratic optimization and so on.People do not only satisfy on the theoretical conclusions about delayed equations, but also concerned practical forms. They hope the solution could be shown intuitionistic. So researchers investigate the numerical solutions by numerical methods, in order to know more about the properties of solutions, see [62] [63] [64].