doi: 10.3934/dcdsb.2022041
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On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production

1. 

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

2. 

College of Mathematics and Statistics, Yunnan University, Kunming 650091, China

*Corresponding author: Pan Zheng

Received  September 2021 Revised  January 2022 Early access March 2022

This paper deals with a quasilinear chemotaxis system with nonlinear signal production
$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} & u_t = \nabla\cdot(\phi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v), & (x, t)\in \Omega\times (0, \infty), \\ & v_t = \Delta v-v+g(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} $
, where
$ \chi\in \mathbb{R} $
, the nonnegative nonlinearities
$ \phi, \psi $
and
$ g $
belong to
$ C^{2}([0, \infty)) $
and satisfy
$ \phi(u)\geq K_{0}(u+1)^{m}, \psi(u)\leq K_{1}u(u+1)^{\alpha-1} $
and
$ g(u)\leq K_{2}(u+1)^{\beta} $
with some
$ K_{0}, K_{1}, K_{2}, \beta>0 $
and
$ \alpha, m\in\mathbb{R} $
.
$ \bullet $
In the chemo-attractive setting, i.e.
$ \chi>0 $
, assume that
$ n\geq1 $
and
$ \beta>1 $
, it is shown that the solution of the above system is global and uniformly bounded provided that
$ \alpha+\beta-m<1+\dfrac{2}{n} $
and
$ m >-\dfrac{2}{n} $
.
$ \bullet $
In the chemo-repulsive setting, i.e.
$ \chi<0 $
, assume that
$ n\geq3 $
and
$ g'(u) \geq0 $
, it is proved that the solution of the above system is also global and uniformly bounded if
$ \alpha-m+\dfrac{n-2}{n+2}\beta<1 $
.
Citation: Runlin Hu, Pan Zheng. On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022041
References:
[1]

N. D. Alikakos, $L^{p}$-bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

N. BellomoA. BellouquidY. 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.

[3]

J. BurczakT. Cie$\acute{s}$lak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal. Theory Methods Appl., 75 (2012), 5215-5228.  doi: 10.1016/j.na.2012.04.038.

[4]

T. Cie$\acute{s}$lak and C. Stinner, Finite-time blow-up and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[5]

T. Cie$\acute{s}$lak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[6]

T. Cie$\acute{s}$lak 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.

[7]

M. Ding and M. Winkler, Small-density solutions in Keller-Segel systems involving rapidly decaying diffusivities, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 47, 18 pp. doi: 10.1007/s00030-021-00709-4.

[8]

M. Freitag, Global existence and boundedness in a chemorepulsion system with superlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A., 38 (2018), 5943-5961.  doi: 10.3934/dcds.2018258.

[9]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-144.  doi: 10.1002/mana.19981950106.

[10]

G. Hazelbauer, \emph{Taxis and Behavior, Elementary Sensory Systems in Biology, Receptors and Recognition}, Series B, Chapman and Hall, London, 1979.

[11]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[12]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, Nonlinear Differ. Equ. Appl., 8 (2001), 399-423.  doi: 10.1007/PL00001455.

[13]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[14]

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.

[15]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[16]

E. Keller and L. 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.

[17]

B. Liu and M. Dong, Global solutions in a quasilinear parabolic-parabolic chemotaxis system with decaying diffusivity and consumption of a chemoattractant, J. Math. Anal. Appl., 467 (2018), 32-44.  doi: 10.1016/j.jmaa.2018.06.001.

[18]

K. Lin and T. Xiang, Strong damping effect of chemo-repulsion prevents blow-up, J. Math. Phys., 62 (2021), Paper No. 041508, 29 pp. doi: 10.1063/5.0032829.

[19]

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.

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funk. Ekva., 40 (1997), 411-433. 

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funk. Ekva., 44 (2001), 441-469. 

[22]

X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math., 359 (2021), 161-168. 

[23]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.

[24]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705.

[25]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.

[26]

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.

[27]

W. Wang and Y. Li, Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production, Nonlinear Anal. Real World Appl., 59 (2021), Paper No. 103237, 21 pp. doi: 10.1016/j.nonrwa.2020.103237.

[28]

W. WangM. Zhang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differ. Equ., 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011.

[29]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.

[30]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[31]

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.

[32]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[33]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[34]

M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[35]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[36]

S. ZhouT. Gong and J. Yang, Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension, Appl. Math. J. Chinese Univ. Ser. B., 35 (2020), 244-252.  doi: 10.1007/s11766-020-3994-5.

show all references

References:
[1]

N. D. Alikakos, $L^{p}$-bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

N. BellomoA. BellouquidY. 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.

[3]

J. BurczakT. Cie$\acute{s}$lak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal. Theory Methods Appl., 75 (2012), 5215-5228.  doi: 10.1016/j.na.2012.04.038.

[4]

T. Cie$\acute{s}$lak and C. Stinner, Finite-time blow-up and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[5]

T. Cie$\acute{s}$lak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[6]

T. Cie$\acute{s}$lak 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.

[7]

M. Ding and M. Winkler, Small-density solutions in Keller-Segel systems involving rapidly decaying diffusivities, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 47, 18 pp. doi: 10.1007/s00030-021-00709-4.

[8]

M. Freitag, Global existence and boundedness in a chemorepulsion system with superlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A., 38 (2018), 5943-5961.  doi: 10.3934/dcds.2018258.

[9]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-144.  doi: 10.1002/mana.19981950106.

[10]

G. Hazelbauer, \emph{Taxis and Behavior, Elementary Sensory Systems in Biology, Receptors and Recognition}, Series B, Chapman and Hall, London, 1979.

[11]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[12]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, Nonlinear Differ. Equ. Appl., 8 (2001), 399-423.  doi: 10.1007/PL00001455.

[13]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[14]

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.

[15]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[16]

E. Keller and L. 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.

[17]

B. Liu and M. Dong, Global solutions in a quasilinear parabolic-parabolic chemotaxis system with decaying diffusivity and consumption of a chemoattractant, J. Math. Anal. Appl., 467 (2018), 32-44.  doi: 10.1016/j.jmaa.2018.06.001.

[18]

K. Lin and T. Xiang, Strong damping effect of chemo-repulsion prevents blow-up, J. Math. Phys., 62 (2021), Paper No. 041508, 29 pp. doi: 10.1063/5.0032829.

[19]

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.

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funk. Ekva., 40 (1997), 411-433. 

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funk. Ekva., 44 (2001), 441-469. 

[22]

X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math., 359 (2021), 161-168. 

[23]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.

[24]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705.

[25]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.

[26]

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.

[27]

W. Wang and Y. Li, Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production, Nonlinear Anal. Real World Appl., 59 (2021), Paper No. 103237, 21 pp. doi: 10.1016/j.nonrwa.2020.103237.

[28]

W. WangM. Zhang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differ. Equ., 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011.

[29]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.

[30]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[31]

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.

[32]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[33]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[34]

M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[35]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[36]

S. ZhouT. Gong and J. Yang, Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension, Appl. Math. J. Chinese Univ. Ser. B., 35 (2020), 244-252.  doi: 10.1007/s11766-020-3994-5.

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