Boundedness and global asymptotic stability for a parabolic-elliptic-ODE chemotaxis-haptotaxis model with remodeling of non-diffusible attractant

Abstract

In this paper, we take into account the multifarious impacts arising from the intricate interplay among chemotaxis, haptotaxis, sub-logistic growth patterns, and remodeling mechanisms on the global boundedness of solutions within a mathematical model. Initially devised by Chaplain and Lolas (2006) \cite{Chaplain7}, this model stands as a potent instrument, illuminating the complex {dynamics that unfolds between}
cancer cells, matrix-degrading enzymes, and the host tissue during the invasive process of cancer cells into the extracellular matrix. The model, outlined as follows, encapsulates a vast array of biological phenomena:
$$
\left\{\begin{array}{ll}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)-
\xi\nabla\cdot(u\nabla w)+f(u,w),\\
\disp{0=\Delta v- v +u},\quad
\\
\disp{w_t=- vw+\eta w(1-u-w).}\quad\\
\end{array}\right.\quad\mbox{in}~\Omega\subset \mathbb{R}^2.
\eqno(*)
$$
Here, $\Omega$ represents a generic bounded domain with a smooth boundary, while $f(u,w)$ encapsulates the proliferation and death of cancer cells, {processes that are intricately intertwined with competition for space involving the extracellular matrix.}
The constants $\chi>0,$ $\xi>0$, and $\eta>0$ reflect various biological processes with nuance.
A pivotal aspect of our exploration focuses on cell kinetics, which {is} meticulously described by a versatile class of sub-logistic source functions. As illustrative {examples}, we consider source functions such as $u\left(\kappa -w- \frac{\mu u}{\ln^\gamma(u+1)}\right)~\mbox{with}~\gamma \in (0,1)$ and $u\left(1-w-\frac{u}{\ln(\ln(u+e))}\right),$
offering nuanced perspectives into the dynamic growth of cancer cells.

In the context of this system, we establish the existence and boundedness of non-negative solutions to the system, thereby significantly broadening the horizons of previous findings reported in \cite{PangPanuuiig1,Taoss,Taox26216,Taox26,Tao79477}. Regarding the qualitative behavior of solutions, our work uncovers an explicit smallness condition on $w_0$ that ensures the exponential decay of $w$ in the long-time limit, while $u$ and $v$ persist in a specific sense.
Our findings contribute immensely to the comprehension of the complex mechanisms that govern cancer cell invasion, offering invaluable insights that could potentially inform the development of targeted therapeutic interventions.