Traveling Wave Fronts In Reaction Diffusion Equation With Fractional Laplacian And Belousov-Zhabotinskii System

The reaction diffusion equation has many applications in physics and biology,and thus the study of its traveling wave fronts and asymptotic behavior of solutions plays an important role in the field of applied Mathematics.In recent years,the reaction diffusion equation with fractional Laplacian has attracted a great deal of attention.The planar traveling wave fronts for the bistable and combustion case have been well studied.For the Fisher-KPP case,the results include the acceler-ated propagation of solutions.In addition,Belousov-Zhabotinskii reaction diffu-sion system has important applications in chemistry and its planar traveling wave fronts have been fully studied.But in high dimensional space,it is shown that there exist more complex nonplanar traveling wave fronts.Nowadays,the research on nonplanar traveling wave fronts of reaction diffusion equations with fractional Laplacian and Belousov-Zhabotinskii system has made initial progress.The main results include the three-dimensional pyramidal traveling waves of reaction diffu-sion equations with fractional Laplacian and the existence and stability of V-shaped traveling wave fronts of Belousov-Zhabotinskii system.In this thesis,we are go-ing to study the traveling wave fronts for the above two kinds of equations.For bistable reaction-diffusion equation with fractional Laplacian,we study the stabili-ty of one-dimensional traveling wave fronts,the existence,uniqueness and stability of two-dimensional V-shaped traveling wave fronts and the existence of cylindrical symmetric traveling wave fronts.For the Belousov-Zhabotinskii system,we study the existence and stability of the pyramidal traveling wave fronts.This paper will be divided into four parts.In this thesis,we first study the stability of one-dimensional traveling wave fronts of bistable reaction-diffusion equation with fractional Laplacian.We prove the comparison principle in the sense of mild solutions.We also overcome the singularity of integral generated by fractional Laplacian at 0 and construct proper super and subsolutions.Then we obtain the global asymptotic stability of traveling wave fronts by using the classical squeezing method.The third chapter studies the V-shaped traveling wave fronts of the bistable reaction-diffusion equation with fractional Laplacian.Since the fractional Laplacian will produce the algebraic decay on functions,the previous method of using hyper-bolic function to construct the supersolutions is no longer applicable.Thus we use a redefined smoothing curve to construct the suitable supersolutions.After that,we use the regularity theory of reaction-diffusion equation with fractional Laplacian to prove the existence of V-shaped traveling wave fronts.Based on the existence of V-shaped traveling wave fronts,we obtain the stability of this nonplanar traveling wave fronts by using the properties of the initial value and the theory of strongly continuous semigroups.In the fourth part of this thesis,we mainly study the cylindrical symmetric trav-eling wave fronts of the bistable reaction-diffusion equation with fractional Lapla-cian.We first analyze the symmetry of the kernel of fractional Laplacian,and use this conclusion to overcome the limitations of the strong maximum principle.At the same time,we obtain the symmetric and monotone properties of three-dimensional pyramidal traveling wave fronts.Then we construct a sequence of pyramidal trav-eling wave fronts and take the limit for this sequence to obtain the cylindrical sym-metric traveling wave fronts.In the last chapter,we study the existence and stability of pyramidal travel-ing wave fronts of the Belousov-Zhabotinskii reaction-diffusion system.We extend the auxiliary function constructed in the two-dimensional V-shape traveling wave fronts of the Belousov-Zhabotinskii system to the three-dimensional case.Then we construct a suitable supersolution,and obtain the existence of pyramidal traveling wave fronts by using the super and subsolutions.After that,we get the stability of the pyramidal traveling wave fronts by using the comparsion principle.