Stability Of Monostable Traveling Wave Solutions Of Reaction-diffusion Equations With Delay And Without Quasi-monotoncity

In the discussion of traveling wave solutions of reaction-diffusion equations,the stability of traveling wave solutions is of the importance and difficulty.Especially,it is more difficult to investigate the stability of critical monostable waves of reaction-diffusion equations with delay and without quasi-monotonicity for the facts that comparison prin-ciple does not hold for the scarcity of quasi-monotonicity in equations,and thus it is not suitable any longer by using the frequent methods to solve the stability of monos-table travelings under quasi-monotonicity conditions,for example,the weighted energy method combined with comparison principle,squeezing technique,and so on.In addi-tion,for the case of critical speed of monostable waves,it is not easy to get the usual weighted L_w~2energy estimate for their decay behaviors at positive infinity.However,the anti-weighted energy method combined with the continuity method does not need com-parison principle holding and it can also overcome the difficulties of energy estimate.Based on the above facts,this thesis is mainly concerned with the stability of monostable traveling wave solutions for two classes of reaction-diffusion equations with delay and non-quasi-monotonicity.The main results in this thesis are as follows:·The stability of monostable traveling waves of a class of reaction-diffusion system with delay and non-quasi-monotonicity is considered.First,we establish the global ex-istence and uniqueness of solutions to corresponding Cauchy problem,a priori estimate and a local estimate of the solutions of perturbed equations under non-quasi-monotonicity condition.By the weighted-energy method combining the continuity method,we then prove the exponential stability of the monostable traveling waves for non-critical case for the system with delay and non-quasi-monotonicity when the initial perturbation is uni-formly bounded at positive infinity but does not necessarily converge to zero.·The stability of monostable traveling waves for critical case for a class of scalar equations with delay and non-quasi-monotonicity is investigated.Firstly,the global exis-tence and uniqueness of solution to perturbed equations are established,where the initial perturbation can be arbitrarily large.Secondly,by using the anti-weighted energy method,the uniform boundedness of solution to perturbed equation is proved under the so-called small initial perturbation.Finally,on the basis of uniform boundedness,the asymptotic stability of monostable traveling wave solutions for critical case for the scalar equation with delay and non-quasi-monotonicity is further proved.