The Existence And Stability Of Traveling Waves With Transition Layers For Some Cross-Diffusion Systems

This thesis is composed of four parts.In the first part, we investigate the existence of traveling waves with transition layers for a class of quasilinear cross-diffusion competition systems proposed by Shigesada et al. By geometric singular perturbation method, for cross-diffusion rate in the second equation sufficiently large, we show that the system has traveling waves with transition layers connecting two semi-trivial equilibrium points and the waves have locally unique slow speed. While for cross-diffusion rate in the second equation sufficiently small, under different assumptions on parameters, the system has traveling waves with transition layers connecting two semi-positive equilibrium points or join semi-positive equilibrium point and positive equilibrium point respectively and the waves have infinitely many wave speeds.The second part is concerned with the stability of traveling wave solutions with unique slow speed obtained in the first part. By spectral analysis method and analytic semigroup theory, we prove that the traveling wave with transition layer is locally asymptotically stable with shift. It is remarked that due to the appearance of cross-diffusion, the linearized system around the traveling wave is a strongly coupled system, with the more generalized eigenvalue problem. Thus when analyzing the eigenvalue problem for the linearized operator, the classical comparison principle is not valid. And the spectral results for competi- tion system or cooperative system based on comparison principle can't be applied directly. Using topological index method, i.e. the first Chen number method (combining a Evans function and a topological invariant), by detailed spectral analysis and topological analysis (including construction and decomposition of unstable plane-bundles for ε > 0 small, topological equivalence of the sub-bundles with the reduced bundles, uniform boundedness of the eigenvalues and location of the eigenvalues for the reduced problems), we can prove the stability of traveling waves with transition layers.In the third part, we study the stability of traveling waves with infinitely many wave speeds obtained in the first part. By using the idea of construction of positive invariant set used in topological index method, combining analytic semigroup theory, we prove that each wave with non-critical speed is locally exponentially stable to perturbations in some exponentially weighted spaces.In the fourth part, we investigate the stability of traveling waves for a class of nerve axon equations and the stability of impulses for a class of chemotaxis models. By detailed analysis, we obtain some spectral properties of the linearized operators. Based on semigroup theory, combined with Evans function method, some methods to determine the exponential stability of wave front for nerve axon equations and the exponential stability of impulses for some chemotaxis models are given.