Entire Solutions Of Time Periodic Reaction-advection-diffusion Equations

The theory of nonlinear parabolic equations is one of the important topic of the modern mathematics researches. As a kind of typical nonlinear parabolic differential equations, reaction-diffusion equations can be used to explain the natural phenomenon that found in many disciplines, such as the species invasion and growth the spread of infectious diseases in biology, the material concentration and temperature changes before and after combustion in chemical, the heat conduction and liberal boundary problems in physics and so on. This paper is mainly concerned with the entire solutions of reaction-advection-diffusion equation in higher dimensions. Here, the entire solutions are defined in the whole space ?and for all time t ?R. In general, the initial-value solutions of parabolic equations is only half the flow(t ?0) from the perspective of dynamic system, and can’t predicate all the information of solutions in systems. Thus, to study entire solutions in dynamics systems is very necessary and practical significance. In fact, entire solution is a full-flow of the equation, we can know the exact moment of the relevant information of the equations by using entire solutions. In addition, the study of entire solutions can help us to understand equation of the transient dynamics from the perspective of mathematical and determine the structure of the global attractor.In this paper, we mainly considered the entire solutions of reactionadvection-diffusion equation with time periodic bistable nonlinearity in infinite cylinders. By introducing the appropriate auxiliary equations, making use of the exponential decay behavior of time periodic traveling wave solutions at infinity, we will construct appropriate supersolutions and subsolutions, and use the comparison principle to prove the existence of entire solutions which behavior as two traveling wave solutions coming from both directions. More over, we also investigate the uniqueness and stability as well as continuous dependence of entire solutions.Unlike homogeneous space, the lack of symmetry between increasing and decreasing periodic traveling fronts caused by the advection, which affects the construction of subsolutions and supersolutions.