Asymptotic Stability Analysis Of The Solutions For Time-Delayed Reaction-Diffusion Equations

Time-delayed reaction-diffusion equations represent an important biological pop-ulation model,and the asymptotic stability analysis of their solutions has significant theoretical and applied values.For the stability of the traveling wave with the boundary effect,because there ex-ists the boundary layer,the ordinary energy estimates are not established.In order to overcome the boundary layer,we use the space shift technique of traveling wave,and establish the accurate weighted energy estimates.For the noncritical traveling wave,when the equation is monotone,combining with the squeeze technique,we construct the proper sub-super solutions to obtain the global exponential asymptotic stability of trav-eling wave.When the equation does not satisfy the monotone conditions,we adopt the weighted energy estimate to get the local exponential asymptotic stability.For the criti-cal traveling wave,use a function transformation which is called anti-weighted method,and after transforming the perturbation equation,and establish the energy estimates to get the stability of the critical traveling wave.For time-delayed reaction-diffusion equation with the nonlocal nonlinear term,when the nonlinear term is not monotone,the traveling wave may be oscillating and sub-super solution method loses effectiveness.Moveover,the regular weighted energy method can not be used either.We adopt the anti-weighted method,and after transforming the perturbation equation,establish the accurate energy estimates to obtain the stability of traveling wave.For the initial-boundary value problem for nonlocal diffusion Nicholson blowflies equation,the solution does not have the higher regularity due to the existence of nonlocal operator.For Dirichlet boundary value,analyse the properties of the principal eigenvalue and obtain the existence and uniqueness of the steady state solution by sub-super solution method.And then,when the equation satisfies the monotone condition,we construct the suitable sub-super solution and gain the solution to convergent to the steady state.For Neumann boundary value,by making sub-super solution sequences,give the asymptotic stability of solutions.