Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Global solvability to a cancer invasion model with remodeling of ECM and porous medium diffusion

Pages: 1493 – 1516

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n6.a1

Authors

Danqing Zhang (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Chunhua Jin (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Abstract

In this paper, we deal with a cancer invasion model with remodeling of ECM and slow diffusion. We consider this problem in a bounded domain of $\mathbb{R}^N (N=2,3)$ with zero-flux boundary conditions, and it is shown that for any large initial datum, the problem admits a global ‘very’ weak solution for any slow diffusion case. It is worth noting that the coexistence of the nonlinear diffusion, haptotaxis and the remodeling of ECM brings essential difficulties. Firstly, unlike the linear diffusion case, the haptotaxis term cannot be merged into the diffusion term, which makes the regularity of ECM less important in the process of making energy estimates. Secondly, the regularity of ECM depends on the worst one of cells density and uPA, therefore, the difficulty caused by the haptotactic term is really highlighted due to the low regularity of ECM. Therefore, it is hard to get the boundedness of cells density because the regularity of ECM is difficult to improve, even for large $m$.

Keywords

‘very’ weak solution, slow diffusion, remodeling mechanism

2010 Mathematics Subject Classification

35K55, 35M10, 92C17

Received 16 March 2021

Received revised 23 November 2021

Accepted 3 January 2022

Published 14 September 2022