On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Xu Pan; Liangchen Wang

doi:10.3934/dcdsb.2021047

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January 2022, 27(1): 361-391. doi: 10.3934/dcdsb.2021047

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Xu Pan and Liangchen Wang ^,

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

^* Corresponding author: Liangchen Wang

Received August 2020 Revised January 2021 Published January 2022 Early access February 2021

This paper deals with the following two-species chemotaxis system with nonlinear diffusion, sensitivity, signal secretion and (without or with) logistic source

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \nabla \cdot (D_1(u)\nabla u - S_1(u)\nabla v) + f_{1}(u),\quad &x\in\Omega,\quad t>0,\\ v_t = \Delta v-v+g_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = \nabla \cdot (D_2(w)\nabla w - S_2(w)\nabla z) + f_{2}(w),\quad &x\in \Omega,\quad t>0,\\ z_t = \Delta z-z+g_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*} $

under homogeneous Neumann boundary conditions in a bounded domain

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

with

$ n\geq2 $

. The diffusion functions

$ D_{i}(s) \in C^{2}([0,\infty)) $

and the chemotactic sensitivity functions

$ S_{i}(s) \in C^{2}([0,\infty)) $

are given by

$ \begin{equation*} \begin{split} D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} \quad \text{and} \quad 0 < S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} \text{ for all } s\geq0, \end{split} \end{equation*} $

where

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

and

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

$ (i = 1,2) $

. The logistic source functions

$ f_{i}(s) \in C^{0}([0,\infty)) $

and the nonlinear signal secretion functions

$ g_{i}(s) \in C^{1}([0,\infty)) $

are given by

$ \begin{equation*} \begin{split} f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} \quad \text{and} \quad g_{i}(s)\leq s^{\gamma_{i}} \text{ for all } s\geq0, \end{split} \end{equation*} $

where

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

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

and

$ k_{i} > 1 $

$ (i = 1,2) $

. With the assumption of proper initial data regularity, the global boundedness of solution is established under the some specific conditions with or without the logistic functions

$ f_{i}(s) $

Moreover, in case

$ r_{i}>0 $

, for the large time behavior of the smooth bounded solution, by constructing the appropriate energy functions, under the conditions

$ \mu_{i} $

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

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

$ t\rightarrow\infty $

Keywords: Boundedness, two-species chemotaxis, two chemicals, logistic source, stabilization.

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

Citation: Xu Pan, Liangchen Wang. On a quasilinear fully parabolic two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 361-391. doi: 10.3934/dcdsb.2021047