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Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy

Part of: Parabolic equations and systems Physiological, cellular and medical topics Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 April 2022

Yifu Wang  and
Chi Xu
Yifu Wang
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China (wangyifu@bit.edu.cn; XuChi1993@126.com)
Chi Xu
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China (wangyifu@bit.edu.cn; XuChi1993@126.com)
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Abstract

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]
with positive parameters $D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$, $\alpha _u,\alpha _w,\mu _u,\beta$. When posed under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{2}$, and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020, J. Differ. Equ. 268, 4973–4997). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate $\beta <1$, the global classical solution $(u,v,w,z)$ is uniformly bounded and exponentially stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $(L^{\infty }(\Omega ))^{4}$ as $t\rightarrow \infty$.

Keywords

HaptotaxisL log L-estimatesasymptotic behaviour

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B45: A priori estimates 35Q92: PDEs in connection with biology and other natural sciences 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 881 - 906
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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