# American Institute of Mathematical Sciences

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

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

 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
 $\left((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{1}}{k_{2}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{2}}{k_{1}-1}}\right)$
as
 $t\rightarrow\infty$
.
Citation: Xu Pan, Liangchen Wang. On a quasilinear fully parabolic two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 361-391. doi: 10.3934/dcdsb.2021047
##### References:
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##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [3] T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar [4] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar [5] T. Cieślak and M. Winkler, Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.  doi: 10.1016/j.nonrwa.2016.10.002.  Google Scholar [6] T. Cieślak and M. Winkler, Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.  doi: 10.1016/j.na.2016.04.013.  Google Scholar [7] M. Ding, W. Wang, S. Zhou and S. Zheng, Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Differential Equations, 268 (2020), 6729-6777.  doi: 10.1016/j.jde.2019.11.052.  Google Scholar [8] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar [9] D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zürich., 2008.  Google Scholar [10] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar [11] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar [12] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [13] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar [14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific., 1996. doi: 10.1142/3302.  Google Scholar [15] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018.  Google Scholar [16] D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388.  doi: 10.1007/s11766-016-3386-z.  Google Scholar [17] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar [18] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.  doi: 10.3934/dcdss.2020015.  Google Scholar [19] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar [20] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar [21] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737.   Google Scholar [22] M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar [23] M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.  Google Scholar [24] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar [25] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in ${\Bbb R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.   Google Scholar [26] X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Mathématique, (2020), to appear. Google Scholar [27] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [28] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  Google Scholar [29] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  Google Scholar [30] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [31] X. Tao, S. Zhou and M. Ding, Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production, J. Math. Anal. Appl., 474 (2019), 733-747.  doi: 10.1016/j.jmaa.2019.01.076.  Google Scholar [32] Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar [33] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar [34] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar [35] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [36] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [37] X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. 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