Cross-diffusion Equations Traveling Wave Solutions Of The Existence Of A Class Of Equations Pulse Stability Of Solution

This thesis is composed of two 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 .Forif b₁/b₂1/a₂1/c₂,by geometric singular perturbation method, for cross-diffusion rateγ2 in the second equation sufficiently large, there exists traveling waves with transition layers connecting two semi-trivial equilibrium points (0,a₂/c₂) and (a₁/b₁,0) and there exists locally unique slow speed .In the second part,we study the stability of the impulseφ= (φ₁,φ₂)for the following equationswhere u₁ andu₂ could be scalar or vector and A is a second order constant coefficient differential operator which generates an analytic semigroup.In the moving coordinate system, the exponential stability of the equations atφwhich equivalent to the exponential stability of the related linear system at dφ/dy is proved at first. Then by detailed analysis,the necessary and sufficient conditions for the exponential stability of the linear system at dφ/dy are obtained.