The Asymptotic Periodic Solutions And The Traveling Wave Solutions For The Reaction-Diffusion Equations

In the study of the applied mathematics, it is very important to reveal the asymptoticbehavior of the solutions for the differential equations and many researchershave been attracted to do so. For example, it is essential to know the evolution ofa population model as time goes on, since it is related to the serious problem of thespecies—to survive or to extinct. Usually, in order to reveal the asymptotic behaviorof the solution for a reaction-diffusion equation, it needs to prove the existence of thesteady-state solution or the periodic solution for the boundary value problem, yet it isimpossible for a nonperiodic time-varying system. Enlightened by the"weighted periodic"phenomenon, we consider these problems by another approach, that is, to studya particular asymptotic behavior—"asymptotic weighted periodicity". In addition, wehave also considered the existence of the wavefront solution (usually called travellingwave solution) which has special asymptotic behavior for the reaction-diffusion equationin a multidimensional cylinder.For a population of a species, its density is affected not only by the environmentbut also by the time delays. In fact, there are so many factors may bring on time delaysto the evolution of a population, such as the hatch period of the Aves, the gestationperiod of the mammals and the retarded supply of the food. What's the asymptoticbehavior of a system with time delays? What about the effects of the time delays ona certain ecosystem? To answer these questions, we give a detailed discussion respectto the food-limited model and the competition model.It is known that a function f(t) is periodic if it satisfies f(t + T) = f(t) for somepositive constant T. This kind of functions accords with the ideal movements of thesubstance in nature. Yet it is not always the case, such as the familiar damp vibrations.We take the swing of a single pendulum as an example. Let f(t) be the function ofthe swing angle. It should satisfies w(t) = f(t + T)/f(t) ≡1 due to the resistanceof the air. In fact, w(t) satisfies 0 < w(t) < 1 in this case. We are enlightened bythis phenomena and have developd a new concept of the"weighted periodicity"in order to enlarge the concept of the"periodicity"for more applications. As our study isjust at the primary stage, here we only consider some of the problems in the ordinarydifferential equations, impulsive differential equations and the functional differentialequations. Our results show that not only the weighted periodic coefficients of theequation but also the impulses can bring about asymptotic weighted periodicity.Fisher found for the first time a wavefront solution of a reaction-diffusion equationin 1937, and after that there is a great advancement in this field, especially for the onedimensonalcase. But for the multidimensonal case, the results for the existence of thewavefronts have appeared in recent years. Wu J. H. and Zou X. F. have developed amonotone iteration scheme which is very universal in dealing with the travelling waveproblems for the one-dimensonal case. Owing to this, we have tried the multidimensonalcase and developed a monotone iteration scheme for the travelling wave problemin a multidimensional cylinder, which makes it possible for us to reveal the existenceof the wavefront solutions for some parabolic equations.