# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021092
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## A free boundary problem of the cancer invasion

 School of Mathematical Science, Harbin Engineering University, Harbin, 150001, Heilongjiang, China

Received  November 2020 Revised  January 2021 Early access March 2021

Fund Project: The first author is supported by NSFC grant 11626072

This paper deals with a free boundary problem for the cancer invasion model over a one dimensional habitat in the micro-environment, in which the free boundary represents the spreading front and is caused by tumour cells and acid-mediated. In this problem it is assumed that the tumour cells spread from the given initial region, and the spreading front expands at a speed that is proportional to the tumour cell and acids' population gradient at the front. The main objective is to realize the dynamics/variations of the healthy cells, tumour cells, acid-mediated and the free boundary. We prove a spreading-vanishing dichotomy for this model, namely the tumour cells either successfully spreads to infinity as time tends to infinite at the front, or it fails to establish and dies out in long run while the healthy cells stabilizes at a positive steady-state. The long time behavior of solution and criteria for spreading and vanishing are obtained.

Citation: Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021092
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##### References:
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