Monostable Traveling Wave Solutions Of Nonlocal Diffusion Equations With Delay And Nonlocal Effects In Space

Traveling wave solutions of reaction-diffusion equations are often used to demonstrate the development of many propagation problems in nature,such as the invasion of species,the spread of infectious diseases and so on,and thus the development direction of biological population and the spread trend of infectious diseases can be predicted by them.Therefore,the research on traveling wave solutions of reaction-diffusion equations necessarily promotes the rapid development of various fields involved in natural science.Note that if the equations contain nonlocal diffusion terms and thus they become the Integro-differential equations,which makes some methods to investigate traveling waves for classical Laplace diffusion equations not be valid.Moreover,it is relatively difficult to study the existence and stability of traveling waves for the sake of the nonlocality in space caused by nonlinear term(including integrals),which need to more detailed analysis and integral techniques.Besides,it is necessary to use anti-weighted technique and Fourier transform to establish the decay estimates of solutions to perturbation equations for the different attenuation forms of monostable traveling waves at ±? under the critical wave velocity.Therefore,for the investigation to existence and stability of monostable traveling waves for the nonlocal diffusion equations with spatially nonlocal effect and delay,it not only widens the investigated form and scope for equations,but improves traveling wave theories of delayed reaction-diffusion equations,which has some research value and practical significance.Based on the above facts,this paper is mainly concerned with the existence of monostable traveling waves for a class of mixed diffusion equations with spatio-temporal delay and the stability of monostable wavefronts under critical wave speed for a class of quasi-monotone equations with delay and nonlocal effects in space.The main work is as follows:The existence of monostable traveling waves for a class of mixed diffusion equations with spatio-temporal delay is investigated.Firstly,by Schauder's fixed point theorem with upper-lower solutions,the existence of monostable traveling waves of the equations under non-critical wave speed (c>c*) and quasi-monotone condition is established.Secondly,the existence of monostable traveling waves of the equations under non-critical wave speed(c>c*) and non-quasi-monotone condition is also obtained by combining the idea of constructing auxiliary equations with Schauder's fixed point theorem.Finally,the existence of monostable traveling waves under critical wave speed (c=c*) and (non) quasi-monotone condition are investigated by analysis techniques and limit theory,respectively.The stability of monostable wavefronts for a class of quasi-monotone equations with delay and nonlocal effects in space under the critical wave speed is considered.Firstly,the solution for the initial value problem with different initial datas and the common upper and lower bounds of traveling waves are obtained based on the comparison principle,and thus the upper bound (lower bound) for the solution of the perturbed equation is also found.Secondly,the decay estimate to the upper bound (lower bound) of the solution to the perturbed equation is established by using the anti-weighted technique and Fourier transform.Finally,the conclusion that the solution of the corresponding initial value problem converges to traveling waves is obtained by squeeze technique,that is,the globally algebraic stability of monostable wavefronts under the critical wave speed (c=c*) is established.