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May  2022, 21(5): 1735-1753. doi: 10.3934/cpaa.2022044

## On an exponentially decaying diffusive chemotaxis system with indirect signals

 a. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China b. School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

* Corresponding author

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

Fund Project: The work is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program (Grant Nos: XJ2021042, 2021-005) and Young Hundred Talents Program of CQUPT in 2022-2024

This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption
 $\begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &(x,t)\in \Omega\times (0,\infty), \\ &v_t = \Delta v+h(v,w), &(x,t)\in \Omega\times (0,\infty), \\ &w_t = \Delta w- w+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a smoothly bounded domain
 $\Omega\subset \mathbb{R}^{n}$
,
 $n\geq2$
, where the nonlinear diffusivity
 $D$
and chemosensitivity
 $S$
are supposed to satisfy
 $K_{1}e^{-\beta^{-}s}\leq D(s) \leq K_{2}e^{-\beta^{+}s} \;\;\;{\rm{and}}\;\;\;\frac{D(s)}{S(s)}\geq K_{3}s^{-\alpha}+\gamma,$
with the constants
 $\beta^{-}\geq \beta^{+}>0$
,
 $K_{1},K_{2},K_{3}>0$
and
 $\alpha,\gamma\geq0$
. When
 $h(v,w) = -v+w$
, we study the global existence and boundedness of solutions for the above system provided that
 $\alpha\in[0,\frac{2}{n})$
,
 $\beta^{-}\geq \beta^{+}>\frac{n}{2}$
,
 $\gamma>1$
and the initial mass of
 $u_{0}$
is small enough. Moreover, it is proved that the global bounded solution
 $(u,v,w)$
converges to
 $(\overline{u_{0}},\overline{u_{0}},\overline{u_{0}})$
in the
 $L^{\infty}$
-norm as
 $t\rightarrow \infty$
, where
 $\overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx$
. When
 $h(v,w) = -vw$
, it is shown that this system possesses a unique uniformly bounded classical solution if
 $\alpha\geq0$
,
 $\gamma>0$
and
 $\beta^{-}\geq \beta^{+}>\frac{n}{2}$
. Furthermore, if
 $n = 2$
,
 $\alpha\geq0$
,
 $\gamma\geq0$
, and
 $\beta^{-}\geq \beta^{+}>\varepsilon$
with some
 $\varepsilon>0$
, we only obtain the global existence of solutions for the above system.
Citation: Pan Zheng, Jie Xing. On an exponentially decaying diffusive chemotaxis system with indirect signals. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1735-1753. doi: 10.3934/cpaa.2022044
##### References:
 [1] 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., 9 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [2] T. Cie$\acute{s}$lak and M. Winkler, Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonl. Anal. Real World Appl., 35 (2017), 1-19.  doi: 10.1016/j.nonrwa.2016.10.002. [3] T. Cie$\acute{s}$lak and M. Winkler, Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonl. Anal., 159 (2017), 129-144.  doi: 10.1016/j.na.2016.04.013. [4] M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.  doi: 10.3934/dcdsb.2018328. [5] M. Ding and M. Winkler, Small-density solutions in Keller-Segel systems involving rapidly decaying diffusivities, Nonlinear Differ. Equ. Appl., 28 (2021), 18 pp. doi: 10.1007/s00030-021-00709-4. [6] M. Fuest, Analysis of a chemotaxis model with indirect signal absorption, J. Differ. Equ., 267 (2019), 4778-4806.  doi: 10.1016/j.jde.2019.05.015. [7] T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2019), 183-217.  doi: 10.1007/s00285-008-0201-3. [8] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. [9] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [10] B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] B. Liu and M. Dong, Global solutions in a quasilinear parabolic-parabolic chemotaxis system with decaying diffusivity and consumption of chemoattractant, J. Math. Anal. Appl., 467 (2018), 32-44.  doi: 10.1016/j.jmaa.2018.06.001. [13] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733–737. doi: http://eudml.org/doc/83404. [14] M. M. Porzio and V. Vespri, H$\ddot{o}$lder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [15] S. Strohm, R. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8. [16] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [17] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010. [18] L. Wang, X. Hu, P. Zheng and L. Li, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, Comp. Math. Appl., 74 (2017), 2444-2448.  doi: 10.1016/j.camwa.2017.07.023. [19] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Meth. Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319. [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [21] M. Winkler, Global existence and slow grow-up in a qualinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b. [22] M. Winkler, Global large-data solution in a chemotaxis-(Navier)-Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-352.  doi: 10.1080/03605302.2011.591865. [23] M. Winkler, Does a 'volume-filling' effect always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.

show all references

##### References:
 [1] 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., 9 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [2] T. Cie$\acute{s}$lak and M. Winkler, Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonl. Anal. Real World Appl., 35 (2017), 1-19.  doi: 10.1016/j.nonrwa.2016.10.002. [3] T. Cie$\acute{s}$lak and M. Winkler, Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonl. Anal., 159 (2017), 129-144.  doi: 10.1016/j.na.2016.04.013. [4] M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.  doi: 10.3934/dcdsb.2018328. [5] M. Ding and M. Winkler, Small-density solutions in Keller-Segel systems involving rapidly decaying diffusivities, Nonlinear Differ. Equ. Appl., 28 (2021), 18 pp. doi: 10.1007/s00030-021-00709-4. [6] M. Fuest, Analysis of a chemotaxis model with indirect signal absorption, J. Differ. Equ., 267 (2019), 4778-4806.  doi: 10.1016/j.jde.2019.05.015. [7] T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2019), 183-217.  doi: 10.1007/s00285-008-0201-3. [8] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. [9] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [10] B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] B. Liu and M. Dong, Global solutions in a quasilinear parabolic-parabolic chemotaxis system with decaying diffusivity and consumption of chemoattractant, J. Math. Anal. Appl., 467 (2018), 32-44.  doi: 10.1016/j.jmaa.2018.06.001. [13] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733–737. doi: http://eudml.org/doc/83404. [14] M. M. Porzio and V. Vespri, H$\ddot{o}$lder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [15] S. Strohm, R. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8. [16] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [17] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010. [18] L. Wang, X. Hu, P. Zheng and L. Li, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, Comp. Math. Appl., 74 (2017), 2444-2448.  doi: 10.1016/j.camwa.2017.07.023. [19] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Meth. Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319. [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [21] M. Winkler, Global existence and slow grow-up in a qualinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b. [22] M. Winkler, Global large-data solution in a chemotaxis-(Navier)-Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-352.  doi: 10.1080/03605302.2011.591865. [23] M. Winkler, Does a 'volume-filling' effect always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.
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