Stability Of Monostable Traveling Wave Solutions Of Nonlocal Reaction-diffusion Equations In Space With Delay

Traveling wave solutions of reaction-diffusion equations have the invariance of spatial translation.Therefore it is often used to describe the phenomena and processes of propagation in the objective world,such as the spread of computer network viruses,migration and invasion of biological populations.In the theories of traveling waves,the stability of monostable traveling waves has always been a hot topic.In particular,the stability of critical monostable traveling wave solutions(which is also called monostable traveling waves)with time-delay and spatial nonlocal effects is considered.It is difficult to construct a couple of appropriate super-solution and sub-solution since there is an unstable point in the monostable traveling wave.The comparison principle does not hold and the monotonicity methods are no effect because of the lack of monotonicity in the equations.At the same time,it is not easy to establish the energy estimation of solutions for the occurrence of spatial nonlocal term.Moreover,the asymptotic behaviors of monostable traveling waves under the critical speed lead to the result that the normal weighted energy method can not be directly applied to the case of critical speed.It is needed to be improved and perfected by combining theory with practice for these challenges we referred above for the monostable traveling waves and its stability of reaction-diffusion equations with delay.As result,in the non-quasi-monotonicity case,we mainly investigate the stability of monostable traveling waves for two kinds of nonlocal reaction-diffusion equations in space with delay in this thesis.The main work in this thesis is as follows:(1)In the non-quasi-monotonicity case,we are concerned with the stability of monostable traveling waves for a class of spatially nonlocal scalar equations with delay.On the one hand,for the non-critical speed case and the so-called small initial perturbation,by using the continuity method combining the weighted-energy method,we establish the stability with exponential decay of the monostable traveling waves of the equation when the initial perturbation is uniformly bounded at positive infinity but is not necessarily converge to zero.On the other hand,the methods of continuation method and anti-weighted energy method are applied to solve the stability of monostable traveling waves for the critical speed case.The asymptotic stability of monostable traveling waves of the scalar equation with delay and without quasi-monotonicity for the critical speed case is proved.(2)In the non-quasi-monotonicity case,we consider the stability of monostable traveling waves of a class of spatially nonlocal reaction-diffusion system with delay.Based on the local existence and uniqueness of solutions to the perturbed equations,the global existence and uniqueness of solutions to the corresponding Cauchy problem are first established by continuation methods.Then,the priori estimates of solutions to the perturbed equations are obtained based on the key inequalities.Finally,when the initial perturbation is sufficiently small,the stability with exponential decay of the monostable traveling waves is thus proved for the non-critical speed case by using the weighted-energy method.