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Global existence in a chemotaxis system with singular sensitivity and signal production
On a quasilinear fully parabolic two-species chemotaxis system with two chemicals
School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China |
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \nabla \cdot (D_1(u)\nabla u - S_1(u)\nabla v) + f_{1}(u),\quad &x\in\Omega,\quad t>0,\\ v_t = \Delta v-v+g_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = \nabla \cdot (D_2(w)\nabla w - S_2(w)\nabla z) + f_{2}(w),\quad &x\in \Omega,\quad t>0,\\ z_t = \Delta z-z+g_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset \mathbb{R}^n $ |
$ n\geq2 $ |
$ D_{i}(s) \in C^{2}([0,\infty)) $ |
$ S_{i}(s) \in C^{2}([0,\infty)) $ |
$ \begin{equation*} \begin{split} D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} \quad \text{and} \quad 0 < S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} \text{ for all } s\geq0, \end{split} \end{equation*} $ |
$ C_{d_{i}},C_{s_{i}}>0 $ |
$ \alpha_i,\beta_{i} \in \mathbb{R} $ |
$ (i = 1,2) $ |
$ f_{i}(s) \in C^{0}([0,\infty)) $ |
$ g_{i}(s) \in C^{1}([0,\infty)) $ |
$ \begin{equation*} \begin{split} f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} \quad \text{and} \quad g_{i}(s)\leq s^{\gamma_{i}} \text{ for all } s\geq0, \end{split} \end{equation*} $ |
$ r_{i} \in \mathbb{R} $ |
$ \mu_{i},\gamma_{i} > 0 $ |
$ k_{i} > 1 $ |
$ (i = 1,2) $ |
$ f_{i}(s) $ |
$ r_{i}>0 $ |
$ \mu_{i} $ |
$ \left((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{1}}{k_{2}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{2}}{k_{1}-1}}\right) $ |
$ t\rightarrow\infty $ |
References:
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X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
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., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
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T. Black,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
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X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
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T. Cieślak and M. Winkler,
Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.
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T. Cieślak 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.
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M. Ding, W. Wang, S. Zhou and S. Zheng,
Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Differential Equations, 268 (2020), 6729-6777.
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Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
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S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
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E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
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G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific., 1996.
doi: 10.1142/3302. |
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K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[16] |
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. |
[17] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 2301-2319.
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M. Mizukami,
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.
doi: 10.3934/dcdss.2020015. |
[19] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
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T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
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L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737.
|
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M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[23] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[24] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[25] |
K. Osaki and A. Yagi,
Global existence for a chemotaxis-growth system in ${\Bbb R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.
|
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X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Mathématique, (2020), to appear. |
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X. Pan, L. Wang and J. Zhang,
Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.
doi: 10.1002/mma.6621. |
[28] |
X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020).
doi: 10.1007/s00033-020-1248-2. |
[29] |
G. Ren and B. Liu,
Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.
doi: 10.1016/j.nonrwa.2019.01.017. |
[30] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[31] |
X. Tao, S. Zhou and M. Ding,
Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production, J. Math. Anal. Appl., 474 (2019), 733-747.
doi: 10.1016/j.jmaa.2019.01.076. |
[32] |
Y. Tao and Z.-A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
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Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[34] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[35] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[36] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[37] |
X. Tu, C. Mu, P. Zheng and K. Lin,
Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.
doi: 10.3934/dcds.2018156. |
[38] |
L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705.
doi: 10.1016/j.jmaa.2019.123705. |
[39] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[40] |
L. Wang, C. Mu, X. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[41] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[42] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
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A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
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Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
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Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
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show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
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., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[3] |
T. Black,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[4] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[5] |
T. Cieślak and M. Winkler,
Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.
doi: 10.1016/j.nonrwa.2016.10.002. |
[6] |
T. Cieślak 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, W. Wang, S. Zhou and S. Zheng,
Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Differential Equations, 268 (2020), 6729-6777.
doi: 10.1016/j.jde.2019.11.052. |
[8] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[9] |
D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zürich., 2008. |
[10] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[11] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[12] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[13] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[14] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific., 1996.
doi: 10.1142/3302. |
[15] |
K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[16] |
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. |
[17] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[18] |
M. Mizukami,
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.
doi: 10.3934/dcdss.2020015. |
[19] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[20] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[21] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737.
|
[22] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[23] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[24] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[25] |
K. Osaki and A. Yagi,
Global existence for a chemotaxis-growth system in ${\Bbb R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.
|
[26] |
X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Mathématique, (2020), to appear. |
[27] |
X. Pan, L. Wang and J. Zhang,
Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.
doi: 10.1002/mma.6621. |
[28] |
X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020).
doi: 10.1007/s00033-020-1248-2. |
[29] |
G. Ren and B. Liu,
Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.
doi: 10.1016/j.nonrwa.2019.01.017. |
[30] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[31] |
X. Tao, S. Zhou and M. Ding,
Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production, J. Math. Anal. Appl., 474 (2019), 733-747.
doi: 10.1016/j.jmaa.2019.01.076. |
[32] |
Y. Tao and Z.-A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[33] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[34] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[35] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[36] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[37] |
X. Tu, C. Mu, P. Zheng and K. Lin,
Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.
doi: 10.3934/dcds.2018156. |
[38] |
L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705.
doi: 10.1016/j.jmaa.2019.123705. |
[39] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[40] |
L. Wang, C. Mu, X. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[41] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[42] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[43] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[44] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[45] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[46] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
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