Traveling Wave Solutions Of A Generalized Diffusive Epidemic Model

Nonlinear funct.ional analysis is based on various nonlinear problems in mathematics,physics,chemistry,biology,medicine,astronomy,cybernetics,engineering,economics and other disciplines.It has established a number of general theories dealing with nonlinear problems,and has gradually become a research direction with profound theoretical significance and wide applica-tion value in modern mathematics.At present,the main contents of non-linear functional analysis include topological degree theory,critical point the-ory,semi-ordering method,analytical met,hod and monotone mapping theory.Topological degree theory is a powerful tool to study qualitative theory of non-linear operatorsFrom it,many famous fixed point theorems can be derived.This paper mainly proves the existence of solutions by Schauder fixed point theorem.The specific contents are as follows:The first chapter is the introduction.The first section mainly introduces the research background.Based on the background,the diffusive epidemic model of two populations with criss-cross mechanism is generalized,and a more general system is obtained as follows:(?)for x ?Rn,ui= ui(x,t)(i = 1,2,3,4),u =(u1,u2,u3,u4),and all parameters are greate.r than 0.At the same time,the hypothesis needed in this paper are given.Section 2 mainly int,roduces the basic knowledge needed in this paper.In Chapter 2,the construction and proof of upper and lower solutions and completely continuous operators are given.Firstly,based on the eigenvalue method,the existence theorem of minimal wave speed c*is given.Secondly,the a.uxiliary system about the original system is introduced.When c>c*,four pairs of upper and lower solutions of the auxiliary systeIIm are constructed.Finally,the set of traveling wave solutions is defined by these four pairs of upper and lower solutions,and the completely continuous operator required by Schauder fixed point theorem is defined on the set.Irn Chapter 3,we prove the existence of strong traveling wave solutions ancd the nonexistence of weak traveling wave solutions.Firstly,we use Schaud?er fixed point t.heorem t.o prove the exist.ence of weak t.raveling wave solutions for the auxiliary system when conditions(C1)or(C2)hold and t.he wave speed satisfies c>c*.That is.when t = ?,the traveling wave.solution connects the equilibrium solution of the initial disease-free state,but,when t =+?,the traveling wave solution does not,connect the equilibrium solution of the final disease-free stat.e.Therefore,we can put forward some reasonable assumptions to ensure the existence of strong traveling wave solutions for auxiliary system-s.Then,by the limit theory and Arzela-Ascoli theorem,we can obtain the existence of strong traveling wave solutions for the original system.Finally,by the two-sided Laplace transform method,we establish the nonexistence of weak traveling wave solution when the conditions(C1)or(C2)are valid and the wave speed satisfies 0<c<c*,or when the condition(C3)is valid and the wave speed satisfies c>0.Chapter 4 is the application and summary.We apply the above conclu-sions to a diffusive influenza disease model with vaccination[13]and a dif-fusive epidemic model with criss-cross mechanism mentioned in the research background(1.1).Finally,we summarize this paper and introduce the short-comings of this paper and future research directions.