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July  2022, 27(7): 4007-4022. doi: 10.3934/dcdsb.2021216

Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

Jianing Xie ,

School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics, Dalian, 116025, China

* Corresponding author: Jianing Xie

Received  March 2021 Revised  July 2021 Published  July 2022 Early access  September 2021

Fund Project: The author is supported by National Science Foundation of China grant (No. 11572081), the Program Funded by Liaoning Province Education Administration (No. LN2021M37) and Dongbei University of Finance and Economics (No. DUFE2020Y20)

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype
$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t>0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} (*)$
in a smoothly bounded domain
$ \Omega\subset\mathbb{R}^N(N\geq1) $
 under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters
$ \mu $
 as well as
$ \delta $
 and
$ \tau $
 are positive. Based on an new energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever
$ \mu>\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $
and the initial data
$ (u_0,v_0,w_0) $
 are sufficiently regular. Here
$ \lambda_0 $
 is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.
Keywords: Boundedness, chemotaxis, indirect signal production, global existence, logistic source.
Mathematics Subject Classification: Primary: 92C17, 35K55, 35K59, 35K20.
Citation: Jianing Xie. Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4007-4022. doi: 10.3934/dcdsb.2021216
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

N. Bellomo, A. Belloquid, 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.

[3]

V. Calvez and J. A. Carrillo, Volume effects in the Keller–Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.

[4]

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.

[5]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 281-301.  doi: 10.1006/aama.2001.0721.

[6]

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.

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Diff. Eqns., 58 (2019), 58-109.  doi: 10.1007/s00526-019-1568-2.

[9]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[10]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. RWA., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.

[11]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.

[12]

H. Matthias and P. Jan, Heat kernels and maximal $L^p$-$L^q$ estimate for parabolic evolution equations, Comm. Partial Diff. Eqns., 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.

[13]

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

[14]

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.

[15]

Q. Tang, Q. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math Sci., 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[16]

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.

[17]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[18]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[19]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[20]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[21]

L. Wang, Y. Li and C. Mu, Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[22]

L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[23]

Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.  doi: 10.1016/j.jde.2015.09.051.

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[25]

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

[26]

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.

[27]

M. Winkler, Global asymptotic stability of constant equilibriain a fully parabolic chemotaxis system with strong logistic dampening, J. Diff. Eqns., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[28]

J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

[29]

J. Zheng, An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion, J. Diff. Eqns., 267 (2019), 2385-2415.  doi: 10.1016/j.jde.2019.03.013.

[30]

J. Zheng, Mathematical research for models which is related to chemotaxis system, current trends in mathematical analysis and its interdisciplinary applications, Birkhäuser, Cham, (2019), 351–444.

[31]

J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi–linear chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 97 (2017), 414-421.  doi: 10.1002/zamm.201600166.

[32]

J. Zheng, Y. Li, G. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns., 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

N. Bellomo, A. Belloquid, 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.

[3]

V. Calvez and J. A. Carrillo, Volume effects in the Keller–Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.

[4]

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.

[5]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 281-301.  doi: 10.1006/aama.2001.0721.

[6]

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.

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Diff. Eqns., 58 (2019), 58-109.  doi: 10.1007/s00526-019-1568-2.

[9]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[10]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. RWA., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.

[11]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.

[12]

H. Matthias and P. Jan, Heat kernels and maximal $L^p$-$L^q$ estimate for parabolic evolution equations, Comm. Partial Diff. Eqns., 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.

[13]

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

[14]

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.

[15]

Q. Tang, Q. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math Sci., 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[16]

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.

[17]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[18]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[19]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[20]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[21]

L. Wang, Y. Li and C. Mu, Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[22]

L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[23]

Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.  doi: 10.1016/j.jde.2015.09.051.

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[25]

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

[26]

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.

[27]

M. Winkler, Global asymptotic stability of constant equilibriain a fully parabolic chemotaxis system with strong logistic dampening, J. Diff. Eqns., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[28]

J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

[29]

J. Zheng, An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion, J. Diff. Eqns., 267 (2019), 2385-2415.  doi: 10.1016/j.jde.2019.03.013.

[30]

J. Zheng, Mathematical research for models which is related to chemotaxis system, current trends in mathematical analysis and its interdisciplinary applications, Birkhäuser, Cham, (2019), 351–444.

[31]

J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi–linear chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 97 (2017), 414-421.  doi: 10.1002/zamm.201600166.

[32]

J. Zheng, Y. Li, G. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064.

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