Version 1
: Received: 18 July 2024 / Approved: 19 July 2024 / Online: 19 July 2024 (09:45:45 CEST)
How to cite:
Ai, B.; Jia, Z. Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production. Preprints2024, 2024071588. https://doi.org/10.20944/preprints202407.1588.v1
Ai, B.; Jia, Z. Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production. Preprints 2024, 2024071588. https://doi.org/10.20944/preprints202407.1588.v1
Ai, B.; Jia, Z. Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production. Preprints2024, 2024071588. https://doi.org/10.20944/preprints202407.1588.v1
APA Style
Ai, B., & Jia, Z. (2024). Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production. Preprints. https://doi.org/10.20944/preprints202407.1588.v1
Chicago/Turabian Style
Ai, B. and Zhe Jia. 2024 "Global Existence and Boundedness of Solutions to a Chemotaxis-Haptotaxis Model With Nonlinear Diffusion and Signal Production" Preprints. https://doi.org/10.20944/preprints202407.1588.v1
Abstract
In this paper, we investigate the following chemotaxis-haptotaxis model
$$\left\{
\begin{array}{ll}
u_{t}=\nabla\cdot (D(u) \nabla u)- \nabla\cdot (H(u)\nabla v)- \nabla\cdot (I(u)\nabla w)+ u(a-\mu u^{k-1}-\lambda w),\\
v_{t}=\triangle v-v+u^{\gamma},\\
w_{t}=-v w
\end{array}\right.\eqno(*)
$$
under homogenous Neumann boundary condition and for a bounded domain $ \Omega \subset \mathbb{R}^{n} (n\geq2)$, with $\lambda, \mu, \gamma >0$, $k>1$, $a \in \mathbb{R}$, and $D(u)\geq K_{D} (u+1)^{m-1}$, $0\leq H(u)\leq \chi u(u+1)^{-\alpha}$, $0\leq I(u)\leq \xi u(u+1)^{-\beta}$ for $K_{D}, \chi, \xi>0, m, \alpha, \beta\in \mathbb{R}$. It has been demonstrated that
(i) For $0\gamma-k+1$ and $\beta>1-k$, problem ($*$) admits a classical solution $(u, v, w)$ which is globally bounded.
(ii) For $\frac{2}{n}\gamma-k+\frac{1}{e}+1$ and $\beta>\max\{\frac{(n\gamma-2)(n\gamma+2k-2)}{2n}-k+1, \frac{(n\gamma-2)(\gamma+\frac{1}{e})}{n}-k+1\}$ or $\alpha>\gamma-k+1$ and $\beta>\max\{\frac{(n\gamma-2)(n\gamma+2k-2)}{2n}-k+1, \frac{(n\gamma-2)(\alpha+k-1)}{n}-k+1\}$, problem ($*$) admits a classical solution $(u, v, w)$ which is globally bounded.
Keywords
Boundedness; Chemotaxis-haptotaxis; Nonlinear diffusion; Signal production
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
