Analysis of long‐time solution of chemotactic model with indirect signal consumption in three‐dimensional case. (16th September 2022)

• Record Type: Journal Article
• Authors: Hu, Qiaoling
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 4(2023)
• Page Start: 3758
• Page End: 3782
• Publication Date: 2022-09-16
• Subjects: boundedness -- eventual smoothness -- global existence -- indirect signal consumption; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8720
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we consider the following chemotaxis model with indirect signal consumption: u t = Δ u m − ∇ · ( u ∇ v ), x ∈ Ω, t > 0, v t = Δ v − v w, x ∈ Ω, t > 0, w t = − δ w + u, x ∈ Ω, t > 0 $$ \left\{\begin{array}{ccc}\hfill {u}_t& =\Delta {u}^m-\nabla \cdotp \left(u\nabla v\right), \hfill & \hfill \kern2em x\in \Omega, t>0, \\ {}\hfill {v}_t& =\Delta v- vw, \hfill & \hfill \kern2em x\in \Omega, t>0, \\ {}\hfill {w}_t& =-\delta w+u, \hfill & \hfill \kern2em x\in \Omega, t>0\end{array}\right. $$ under the homogeneous Neumann boundary conditions for u $$ u $$ and v $$ v $$ in a bounded convex domain Ω ⊂ ℝ 3 $$ \Omega \subset {\mathbb{R}}^3 $$ with smooth boundary, where δ > 0 $$ \delta >0 $$ is a given parameter. It is shown that for each m > 4 / 3 $$ m>4/3 $$, the global weak solutions exist whenever the initial data ( u 0, v 0, w 0 ) $$ \left({u}_0, {v}_0, {w}_0\right) $$ are sufficiently regular satisfying u 0 ≥ 0, v 0 ≥ 0 $$ {u}_0\ge 0, {v}_0\ge 0 $$ and w 0 ≥ 0 $$ {w}_0\ge 0 $$ . Moreover, it is proved that the global solution ( u, v, w ) $$ \left(u, v, w\right) $$ converges to ū 0, 0, ū 0 δ $$ \left({\bar{u}}_0, 0, \frac{{\bar{u}}_0}{\delta}\right) $$ in L ∞ ( Ω ) $$ {L}^{\infty}\left(\Omega \right) $$ as t → ∞ $$ t\to \infty $$ . In particular, we extend the recent result of Liu–Li–Huang [Y Liu, Z Li and J Huang, Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption, J. Differential Equations 269 (2020) 6365‐6399] to the nonlinear diffusion case.