Bistable Waves For Two Kinds Of Nonlocal Diffusion Equations With Delay

As one of the main research topics in reaction-diffusion equations,many propagation problems in the fields of ecology,epidemiology and population dynamics,such as the spread of infectious diseases and the invasion of species can be described by traveling wave solutions,and these problems can be solved by investigating the existence,uniqueness and stability of traveling waves of the corresponding models,in which monostable traveling waves and bistable traveling waves are mainly research types.Compared with monostable traveling waves,it is relatively difficult to investigate because there also exists an unstable equilibrium between two stable equilibria for bistable waves,and thus it is necessary to rule out the two cases(one case is traveling wave solutions tend to the left side of stable equilibrium and the middle of unstable equilibrium at at the positive infinity and negative infinity,respectively,and the other is traveling wave solutions tend to the right side of stable equilibrium and the middle of unstable equilibrium at at the positive infinity and negative infinity,respectively)when we establish the existence of bistable waves by using the homotopy technology.In addition,the emergence of mixed diffusion makes the estimations for wave speeds and waveprofiles more complicated.Fortunately,the difficulty can be overcome by priori estimations and the detailed analysis techniques.On the other hand,it is also worth investigating the stability of bistable waves for nonlocal diffusion systems with delay.Notice that when we are concerned with the global stability of bistable waves by using the convergence theory of monotone semiflows,it is hard to obtain the conclusion that the image orbit to the monotone semiflow of solutions is a precompact set corresponding to the initial value problem for scalar equations or systems due to the appearance of the time delay.However,the squeezing technique based on comparison principle and upper and lower solutions is a powerful tool to establish the global stability of bistable waves and it is no need to testify the relative compactness for the semiflow of solutions,although it makes the construction of upper and lower solutions to equations complicated for the sake of time delay and the coupling from different functions in the system.Based on these facts,the existence and stability of bistable waves for two classes of nonlocal diffusion equations with delay are investigated in this paper.The specific research work is as follows:The existence and uniqueness of bistable waves for a class of reaction-diffusion-convection equations with mixed diffusion and delay are investigated.The existence of bistable waves is based on homotopy technique.Firstly,a family of continuously parameterized equations is constructed based on the traveling wave equation of original equation.Secondly,the existence of bistable waves for the parameterized equations with ??[0,1)is obtained by implicit function theorem.Finally,the existence of bistable waves for the original equation is verified by the convergence of subsequence when ?=1.In addition,based on the existence and monotonicity of bistable waves,the uniqueness(up to translation)of bistable wavefronts for the equation is established by the sliding technique.The stability of bistable wavefronts for a class of epidemic system with nonlocal diffusion and delay is considered.Firstly,the existence,uniqueness and comparison principle of solutions to the initial value problem corresponding to the original system are established.Secondly,an iteration lemma and a final estimation of the solution to the initial value problem are obtained by using the super-and subsolutions and comparison principle.Finally,the globally exponential stability of bistable wavefronts of the system is established by using squeezing technique.Furthermore,Lyapunov stability and uniqueness of bistable wavefronts are also obtained.