Nonlinear Stability Of Traveling Waves To Singular Chemotactic Models

In this thesis,we study the stability of traveling waves to a PDE-ODE chemotaxis model with general consumption function.This model describes the phenomenon of the tumor invasion to healthy tissue.The mathematical feature of the model is that,it has a logarithmic sensitive function,and the traveling wave has a vacuum end state,which cause singularity.Thus it is very challenging to study the stability of traveling waves.We first employ the Cole-Hopf transformation to transform the original PDEODE model into a parabolic-hyperbolic system.And then using the theory of viscous conservation law,which in combination with the weighted energy method,we overcome the difficulties of singularity and vacuum.Finally,we get the nonlinear stability of onedimensional wave and the two-dimensional planar wave.This result extends the work of reference [54] to the cases of one dimensional traveling waves with vacuum and two dimensional planar waves.