Early and late stage profiles for a new chemotaxis model with density-dependent jump probability and quorum-sensing mechanisms

TL;DR

This paper introduces a new chemotaxis model with degenerate diffusion and density-dependent sensitivity, capturing early and late tumor cell migration stages with rigorous speed estimates and convergence results.

Contribution

It develops a novel chemotaxis model with degenerate diffusion and analyzes its early and late stage behaviors, including speed estimates and asymptotic convergence.

Findings

Tumor region expands at rate O(t^β) with 0<β<1/2 during early stage.

The system's solution converges exponentially to the steady state.

The model addresses challenges of degenerate diffusion in chemotaxis.

Abstract

In this paper, we derive a new chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, the new model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole body, 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.