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EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITY-DEPENDENT JUMP PROBABILITY

1. School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China
2. School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China
3. Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada
4. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

In this paper,we derive a chemotaxis model with degenerate diffusionand density-dependent chemotactic sensitivity,and we provide a more realistic description of cell migration processfor its early and late stages.Different from the existingstudies focusing on the case of non-degenerate diffusion, this model withdegenerate diffusion causes us some essential difficulty on the boundedness estimatesand the propagation behavior of its compact support.In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain,we first estimate the expanding speed of tumour region as $O(t^{\beta})$ for $0<\beta<\frac{1}{2}$.Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state.The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.
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