Global boundedness for a <inline-formula><tex-math id="M1">\begin{document}$ \mathit{\boldsymbol{N}} $\end{document}</tex-math></inline-formula>-dimensional two species cancer invasion haptotaxis model with tissue remodeling

Feng Dai; Bin Liu

doi:10.3934/dcdsb.2021044

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January 2022, 27(1): 311-341. doi: 10.3934/dcdsb.2021044

Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling

Feng Dai ^1,2, and Bin Liu ^1,2,,

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China
2.	Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

^* Corresponding author: Bin Liu

Received September 2020 Revised December 2020 Published January 2022 Early access February 2021

Fund Project: This work is supported by National Natural Science Foundation of China grant 11971185

This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling

$ \begin{equation} \begin{cases} c_{1t} = \Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{\rm EMT}c_1+\mu_1c_1(r_1-c_1^\kappa-c_2-v),\\ c_{2t} = \Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{\rm EMT}c_1+\mu_2c_2(r_2-c_1-c_2^\kappa-v),\\ \tau m_t = \Delta m+c_1+c_2-m,\\ v_t = -mv+\eta v(1-c_1-c_2-v) \end{cases}\nonumber \end{equation} $

in a bounded and smooth domain

$ \Omega\subset\mathbb{R}^N\;(N\geq1) $

with zero-flux boundary conditions for

$ c_1,c_2 $

and

$ m $

, where

$ \chi_i,\mu_i,r_i>0\;(i = 1,2) $

$ \eta>0 $

$ \kappa\geq1 $

$ \tau\in\{0,1\} $

, and

$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $

is the epithelial-mesenchymal transition rate function such that

$ \mu_{\rm EMT}\leq\mu_M $

with some constant

$ \mu_M>0 $

. When

$ \kappa = 1 $

and

$ N = 3 $

, by rasing the coupled a priori estimates of

$ c_1 $

and

$ c_2 $

in the following way

$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $

with any

$ p>2 $

, it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small

$ r_i $

and

$ \mu_M $

. When

$ \kappa>1 $

and

$ N\geq1 $

, by rasing the coupled a priori estimates of

$ c_1 $

and

$ c_2 $

from

$ L^1(\Omega) $

$ L^p(\Omega) $

with any

$ p>1 $

, then to

$ L^\infty(\Omega) $

, it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary

$ r_i $

and

$ \mu_M $

. The result for

$ \kappa = 1 $

complements previously known one, and the result for

$ \kappa>1 $

is new.

Keywords: Haptotaxis, cancer invasion, global boundedness, tissue remodeling, a priori estimates.

Mathematics Subject Classification: Primary: 35K55; Secondary: 35A01, 35A09, 35B45, 92C17.

Citation: Feng Dai, Bin Liu. Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 311-341. doi: 10.3934/dcdsb.2021044