Boundedness And Rates Of Convergence In A Quasilinear Chemotaxis Model For Tumor Invasion

The chemotaxis model for tumor invasion is a extended model with classic chemotaxis model.Tumor invasion contains many important steps involved different biological mechanisms.Many biologists and mathematicians established various mathematic models for tumor invasion,especially a new tumor invasion model originally introduced by Chaplain and Anderson.This paper deals with the well-posedness problem to the above system which describes the invasion of tumor cells.This paper divides into four sections.In Section 1,we mainly introduce the background of chemotaxis system and the invasion of tumor cells and introduce the research status on the chemotaxis model for tumor invasion at home and abroad in details.In Section 2,we consider the local existence and global existence for the solutions to the chemotaxis system for tumor invasion which consists of nonlinear diffusion function.This model includes three parabolic equations and one ordinary differential equation and four variables?u,v,w,z?.Relying on heat semigroup techniques,we first obtain some boundedness for the component z in some suitable L^p????p>1?space through the boundedness of ?u?L¹???.Then it is obvious to obtain some estimates for ?v in L^q????q>1?spaces.Therefore,when the exponent of diffusion function greater than some numbers,we obtain the uniform boundedness of u in L^????.In Section 3,we investigate the large time behavior of the solutions to above system.Making use of the energy function which decreases as time increases,we first discuss the large time behavior of w and v in W^1,????.Moreover,when given a lower bound for the diffusion function,we prove that the long time behaviors of u and z in L^???? and W^1,????,respectively.In Section 4,we summarize the main conclusions,and propose some interesting problems to be done.