Propagation Dynamics Of Bistable Reaction-diffusion Equations In Cylinders With Dirichlet Boundary Condition

This dissertation is concerned with bistable traveling waves for reaction-diffusion equations(RDEs)in cylinders subject to Dirichlet boundary condition,the content of which includes:(1)the bistability structure and the existence of bistable traveling waves(BTWs)of time-periodic RDEs;(2)the bistability structure and the existence,uniqueness(up to a translation),global exponential asymptotic stability of BTWs of ignition-type RDEs.First,to generalize existing theories of BTWs for monotone semiflows,by replacing the positive cone of phase space in the original theories by an appropriate subcone with a nonempty interior and following the same ideas therein,we prove an extension result suitable for cylindrical RDEs with Dirichlet boundary condition.Secondly,under bistability structure hypothesis(BSH),the existence of BTWs of time-periodic RDEs in strips with Dirichlet boundary condition is verified by virtue of the above extension result.Moreover,for a reaction term with periodic perturbation,several sufficient conditions for the corresponding autonomous system to own a bistability structure are established in the absence of periodic perturbation.When the periodic perturbation is sufficiently small,a perturbation argument is employed to confirm the bistable structure of the periodic system.Lastly,suppose that BSH holds,by virtue of fundamental solutions and operator semigroup theories,the solution semiflow of ignition-type RDEs in cylinders with Dirichlet boundary condition is qualified for all the hypotheses of the generalized theories of monotone semiflows,from which the existence of BTWs follows.To establish sufficient conditions for the ignition system to own a bistable structure,bifurcation theories are utilized to obtain the global bifurcation diagram of positive solutions of the relevant elliptic problem,then the bistability structure of the ignition system is justified as well.Moreover,the global exponential asymptotic stability and uniqueness of BTWs are proved by using the squeezing technique and constructing refined upper-lower solutions.