Stability Of Traveling Wave Solutions Of Reaction-Diffusion Equations With Delay

Among the various qualitative properties of traveling wave solutions of reaction-diffusion equations, stability is one of the more important ones. However, it is relatively difficult to handle stability, especially for the stability of traveling wave solutions of (non-local)delayed reaction-diffusion equations with(without) quasi-monotonicity. One reason is that the standard theories and frequently used techniques that can be used to handle the stability of traveling wave solutions of classic reaction-diffusion equations cannot be used again because of the appearance of (nonlocal)delay or couple. For example, when we study the stability of traveling wave solutions for delayed systems without quasi-monotonicity, squeezing technique and the weighted-energy method combining compar-ison principle are no effect because comparison principle does not hold. Therefore, for (nonlocal)delayed reaction-diffusion equations with(without) quasi-monotonicity, the in-vestigation of the stability of traveling wave solutions not only complements some stability methods of traveling wave solutions in theory, but also fills up some gaps in those un-solved stability problems of traveling wave solutions in fact. Based on the above fact, in this thesis, we present our investigations in the stability of mono-bistable waves of some (nonlocal)delayed scalar equation and systems with(without) quasi-monotonicity. The main results in this thesis are as follows.●We consider the stability of traveling fronts of a monostable scalar equation with nonlocal delay by taking two kinds of different delay kernels. By the weighted-energy method combining comparison principle, we establish the global exponential stability of all traveling fronts under the so-called large initial perturbation (i.e. the initial perturbation around the traveling wave decays exponentially as x→-∞, but it can be arbitrarily large in other locations) for the scalar equation, even including the slower waves whose wave speed are close to the critical speed.●We are concerned with the stability of monostable waves of a class of epidemic system with delay. When the system is satisfied with quasi-monotonicity condition, the existence and comparison theorems of solutions in weighted space of corresponding Cauchy problem are first established for the system on R by appealing to the theories of analytic semigroup and abstract functional differential equations, then the methods of weighted-energy method combining comparison principle are applied to solve the stability of monos-table fronts of delayed reaction-diffusion systems with quasi-monotonicity in some appropriate exponential weighted space and prove the global exponential stabil-ity of monostable fronts under the so-called large initial perturbation. Furthermore, when the system is not satisfied with quasi-monotonicity condition, the global existence and uniqueness of solutions of corresponding Cauchy problem are first established for the sys-tem on R, then the methods of weighted-energy method combining continuation method are applied to solve the stability of monostable waves of delayed reaction-diffusion systems without quasi-monotonicity in some appropriate exponential weighted space and prove the exponential stability of monostable waves under the so-called small initial perturbation (i.e. the initial perturbation around the traveling wave is suitable small in a weighted norm). In particular, we study the exponential stability of traveling wave solutions of the delayed epidemic model with crossing-monostable nonlinearity.●We study the bistable waves of a class of delayed systems with quasi-monotonicity. The existence and comparison theorems of solutions of the corresponding Cauchy prob-lem are first established for the delayed systems on R by appealing to the theories of analytic semigroup and abstract functional differential equations. The global exponential stability of bistable fronts is then proved by using squeezing technique which is based on comparison principle and construction of appropriate sub-solution and super-solution. Furthermore, monotonicity, Lyapunov stability and uniqueness of bistable waves are ob-tained too. Finally, as an application of the main results, the bistable waves of a delayed epidemic model satisfying quasi-monotonicity condition are discussed.