Multidimensional Propagation Phenomena Of Reaction-Diffusion Equations With Combustion And Monostable Nonlinearities

Since reaction-diffusion equation can describe many diffusion phenomena in na-ture,it has aroused widespread concern and research.For example,burning flame,the spread of disease,heat conduction phenomena and so on.As a special type of solutions of reaction-diffusion equation,traveling wave solution can explain many propagation phenomena in nature.Then it is one of the most important topics of the modern mathematics research,where planar traveling front have been widely studied.However,duo to the influence of curvature and spatial dimensions,when the traveling wave solution propagates in Rn(n?2),its level set may no longer be a hyperplane.Therefore,studying the non-planar traveling wave solutions of the reaction-diffusion equation can make us fully understand the complex propagation form of traveling wave solutions.On the other hand,whether it is a planar trav-eling wave solution or a non-planar traveling wave solution,they converge to the equilibrium point far away from their moving level sets,uniformly in time.It is precisely because of the concept of transition traveling wave solution based on this essential feature,which allows us to deal with the phenomenon of propagation in more general media and can make us fully study the dynamic behavior of equation.In this thesis,we first study the existence?uniqueness and stability of nonplanar traveling wave solution of reaction-diffusion equations with combustion and monos-table nonlinearities in Rn.Secondly,we study the qualitative and the existence of new type transition fronts of combustion reaction-diffusion equations in Rn.Chapter 1 mainly introduces the research status of traveling wave solution of reaction-diffusion equations and state the research problems of the present thesis.Chapter 2 studies the existence of two dimensional V shaped traveling fronts and three dimensional pyramidal traveling fronts of reaction-diffusion equations with combustion and monostable nonlinearities.By constructing suitable supersolutions and subsolutions,we obtain the existence of V shaped traveling fronts and pyramidal traveling fronts in two dimensional spaces and three dimensional spaces respectively using the comparison principle.In chapter 3,we investigate the stability of two dimensional V shaped traveling fronts of reaction-diffusion equations with combustion and monostable nonlineari-ties.By constructing a series of supersolution and subsolution,using comparison principle,we obtain the global asymptotically stability of two dimensional V shaped traveling fronts in R2.In chapter 4,we study the stability of two dimensional V shaped traveling fronts of reaction-diffusion equations with combustion and monostable nonlineari-ties in high dimensional.Firstly,using super-subsolution method combined with comparison principle,we obtain that the two dimensional V shaped traveling front is stable in Rn(n?3)under the initial perturbations decaying at infinity.In par-ticular,if the initial values are some perturbations of V shaped traveling front in L1?L?,then the V shaped traveling front is algebraically stable in Rn(n?3).Secondly,we prove that V shaped traveling front is unstable under the general bounded perturbations.Chapter 5 studies the stability of three dimensional pyramidal traveling fronts of reaction-diffusion equations with combustion and monostable nonlinearities.Firstly,we characterize the pyramidal traveling front as a combination of planar fronts on lateral surfaces.Secondly,by establishing the derivative estimation and multiple solutions,using sliding plane technology and the squeezing technique,we establish the global stability and uniqueness of pyramidal traveling fronts.Chapter 6 studies the transition fronts of combustion reaction-diffusion equa-tions.Firstly,we show that all transition fronts have a global mean speed and it is unique.Secondly,we give a characterization of planar fronts among the more general class of almost-planar fronts.Thirdly,we prove that the existence of new type transition fronts,which are not invariant as times runs in any moving frame.Finally,we show that all transition fronts are monotone increasing in time t.