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Isochronicity of bi-centers for symmetric quartic differential systems
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 |
$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t>0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} (*)$ |
$ \Omega\subset\mathbb{R}^N(N\geq1) $ |
$ \mu $ |
$ \delta $ |
$ \tau $ |
$ \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. $ |
$ (u_0,v_0,w_0) $ |
$ \lambda_0 $ |
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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