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doi: 10.3934/dcdsb.2021216
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## Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

 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 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.
Citation: Jianing Xie. Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021216
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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