Entire Solutions Of Reaction-Advection-Diffusion Equations In High Dimensions And Applications

The theory of nonlinear parabolic differential equations is an important component of the modern mathematic researches.In this paper,we concerned with the entire solutions of a reaction-advection-diffusion equation in higher dimensions.Here,the entire solutions are defined in the whole space and for all time t∈R.In fact,entire solution is a full-flow of the equation.By using entire solution,we can know the exact moment of the relevant information about the equations.Thus,the study of the entire solutions is necessary and practical significance.Indeed,front propagation occurs in many applied problems,such as chemical kinetics,combustion,transport in porous media and biology.For specific front propagation--traveling wave solution,are special kinds of entire solutions.In addition to traveling wave solutions,the interaction between them is also an important topic in the study of reaction-diffusion equations,which is crucially related to the pattern formation problem,specially to the time evolutional process of localized patterns,where more important information on the evolutional process of patterns are given and there are important application in physical,Chemical,biological,physiological systems.From the dynamical points of view,the study of entire solution is essential for a full understanding of the transient dynamics and the structure of the global attractors.Also,entire solutions can be used to imply that the dynamics of two solutions can have distinct histories in the configuration.By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish entire solutions of nonlinear parabolic differential equations in higher dimensions.The main techniques are to characterize the asymptotic behavior of the traveling wave solutions as t→-∞. Though there are many well-known results of entire solutions of reaction-diffusion equations in the one-dimensional space,the issue of the existence of entire solutions of reactionadvection -diffusion equations which admits nonplanar traveling wave solutions,is still open.Comparing with the case of one-dimensional space,which is related to a second order ordinary differential equations,the case of study entire solutions in higher dimensions actually related to elliptic equations,which is not only more meaningful and valuable in theory and practice,but also more challengeable in mathematics to study such equations. Especially,for the ignition temperature nonlinearity,we need to establish the theory of entire solutions.These are the mainly motivations of this thesis.First,we consider the existence of entire solutions of a reaction-advection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders.A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave fronts coming from both directions.In order to illustrate our main results,a passive-reaction-diffusion equation model arising from propagation of fronts is considered.In next chapter,we deal with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders.Assume that the equation admits three equilibria:two stable equilibria 0 and 1,and an unstable equilibriumθ.It is well known that there are different wave fronts connecting any two of those three equilibria.By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish three different types of entire solutions.In the sequel,we study the uniqueness and Liapunov stability of entire solutions for bistable reaction-advection-diffusion equation in heterogeneous media.By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state,we proved that there exist entire solutions behaving as two traveling curved fronts coming from both directions,and we prove that such an entire solution is unique and is Liapunov stable.At last,we establish the existence of pulsating entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in periodic framework.By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function,we proved that there exist pulsating entire solutions behaving as two pulsating fronts coming from both directions, and approaching each other.