Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production

Xu Pan; Liangchen Wang

doi:10.3934/cpaa.2021064

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June 2021, 20(6): 2211-2236. doi: 10.3934/cpaa.2021064

Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production

Xu Pan and Liangchen Wang ^,

School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

^* Corresponding author

Received October 2020 Revised March 2021 Published June 2021 Early access April 2021

Fund Project: This work is supported by the Chongqing Research and Innovation Project of Graduate Students (No. CYS20271) and the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202000618)

This paper deals with the following quasilinear two-species chemotaxis system

$ \begin{equation*} \begin{cases} \partial_{t} u_1 = \nabla \cdot (D_1(u_{1})\nabla u_{1} - S_1(u_{1})\nabla v) + f_{1}(u_{1}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} u_2 = \nabla \cdot (D_2(u_{2})\nabla u_{2} - S_2(u_{2})\nabla v) + f_{2}(u_{2}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} v = \Delta v-v+g_1(u_{1})+g_2(u_{2}),\quad &x\in\Omega,\quad t>0 \end{cases} \end{equation*} $

under homogeneous Neumann boundary conditions in a bounded domain

$ \Omega\subset \mathbb{R}^{n} $

$ (n\geq2) $

. The diffusivity and the density-dependent sensitivity are given by

$ D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} $

and

$ S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} $

for all

$ s\geq0 $

, respectively, where

$ C_{d_{i}},C_{s_{i}}>0 $

and

$ \alpha_i,\beta_{i} \in \mathbb{R} $

; the logistic source and the signal productions are given by

$ f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} $

and

$ g_{i}(s)\leq s^{\gamma_{i}} $

for all

$ s\geq0 $

respectively, where

$ r_{i} \in \mathbb{R} $

$ \mu_{i},\gamma_{i} > 0 $

and

$ k_{i} > 1 $

$ (i = 1,2) $

. It is proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions

$ f_{i}(s) $

, which partially improves the results in [25]. Moreover, in case

$ r_{i}>0 $

, if

$ \mu_{i} $

are sufficiently large, it is shown that the global bounded solution exponentially converges to

$ ((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{1}}{k_{1}-1}} + (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{2}}{k_{2}-1}}) $

$ t\rightarrow \infty $

Keywords: Boundedness, chemotaxis, nonlinear signal production, quasilinear, stabilization, two-species.

Mathematics Subject Classification: Primary: 92C17, 35K35; Secondary: 35A01, 35B35.

Citation: Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064