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When do Keller–Segel systems with heterogeneous logistic sources admit generalized solutions?
1. | Institute for Applied Mathematics, School of Mathematics, Southeast University, Nanjing 211189, China |
2. | Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany |
$ \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \lambda(x) u - \mu(x) u^\kappa, \\ v_t = \Delta v - v + u \end{cases} $ |
$ \Omega \subset \mathbb R^n $ |
$ n \geq 2 $ |
$ \lambda, \mu $ |
$ \kappa $ |
$ \mu(x) = |x|^\alpha $ |
$ \alpha < 2 $ |
$ \kappa = 2 $ |
$ \mu \equiv \mu_1 > 0 $ |
$ \kappa > \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\} $ |
$ \lambda $ |
$ \kappa > \frac{2n+4}{n+4} $ |
$ n = 2 $ |
$ \kappa > 1 $ |
References:
[1] |
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. |
[2] |
P. Biler, Radially symmetric solutions of a chemotaxis model in the plane–the supercritical case, in Parabolic and Navier–Stokes Equations. Part 1, vol. 81 of Banach center publ., Polish Acad. Sci. Inst. Math., Warsaw, 2008, 31–42.
doi: 10.4064/bc81-0-2. |
[3] |
T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems, Preprint, arXiv: 2005.12089. Google Scholar |
[4] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[5] |
M. Fuest, Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 103022, 14pp.
doi: 10.1016/j.nonrwa.2019.103022. |
[6] |
F. Heihoff, Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term, Z. Für Angew. Math. Phys., 71 (2020), Paper No. 80, 23 pp.
doi: 10.1007/s00033-020-01304-w. |
[7] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
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. |
[9] |
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. |
[10] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
J. Lankeit, Immediate smoothing and global solutions for initial data in ${L}^1 \times {W}^{1, 2}$ in a Keller-Segel system with logistic terms in 2D, Proc. R. Soc. Edinb. Sect. Math, to appear (see also arXiv: 2003.02644). Google Scholar |
[13] |
J. Lankeit and M. Winkler,
Facing low regularity in chemotaxis systems, Jahresber. Dtsch. Math.-Ver., 122 (2020), 35-64.
doi: 10.1365/s13291-019-00210-z. |
[14] |
X. Li,
On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.
doi: 10.1016/j.nonrwa.2019.02.005. |
[15] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[16] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvac, 44 (2001), 441–469. URL http://www.math.kobe-u.ac.jp/ fe/xml/mr1893940.xml. |
[17] |
K. J. Painter,
Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theor. Biol., 481 (2019), 162-182.
doi: 10.1016/j.jtbi.2018.06.019. |
[18] |
R. B. Salako and W. Shen,
Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.
doi: 10.1142/S0218202518400146. |
[19] |
R. B. Salako and W. Shen,
Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. II. Existence, uniqueness, and stability of strictly positive entire solutions, J. Math. Anal. Appl., 464 (2018), 883-910.
doi: 10.1016/j.jmaa.2018.04.034. |
[20] |
R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. III. Transition fronts, Preprint, arXiv: 1811.01525. Google Scholar |
[21] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-368.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[22] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[23] |
G. Viglialoro,
Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.
doi: 10.1016/j.jmaa.2016.02.069. |
[24] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[25] |
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. |
[26] |
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. |
[27] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[28] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Mathématiques Pures Appliquées, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[29] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[30] |
M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Für Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[31] |
M. Winkler,
How strong singularities can be regularized by logistic degradation in the Keller–Segel system?, Ann. Mat. Pura Ed Appl., 198 (2019), 1615-1637.
doi: 10.1007/s10231-019-00834-z. |
[32] |
M. Winkler,
The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in ${L}^1$, Adv. Nonlinear Anal., 9 (2019), 526-566.
doi: 10.1515/anona-2020-0013. |
[33] |
D. Woodward, R. Tyson, M. Myerscough, J. Murray, E. Budrene and H. Berg,
Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189.
doi: 10.1016/S0006-3495(95)80400-5. |
show all references
References:
[1] |
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. |
[2] |
P. Biler, Radially symmetric solutions of a chemotaxis model in the plane–the supercritical case, in Parabolic and Navier–Stokes Equations. Part 1, vol. 81 of Banach center publ., Polish Acad. Sci. Inst. Math., Warsaw, 2008, 31–42.
doi: 10.4064/bc81-0-2. |
[3] |
T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems, Preprint, arXiv: 2005.12089. Google Scholar |
[4] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[5] |
M. Fuest, Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 103022, 14pp.
doi: 10.1016/j.nonrwa.2019.103022. |
[6] |
F. Heihoff, Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term, Z. Für Angew. Math. Phys., 71 (2020), Paper No. 80, 23 pp.
doi: 10.1007/s00033-020-01304-w. |
[7] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
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. |
[9] |
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. |
[10] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
J. Lankeit, Immediate smoothing and global solutions for initial data in ${L}^1 \times {W}^{1, 2}$ in a Keller-Segel system with logistic terms in 2D, Proc. R. Soc. Edinb. Sect. Math, to appear (see also arXiv: 2003.02644). Google Scholar |
[13] |
J. Lankeit and M. Winkler,
Facing low regularity in chemotaxis systems, Jahresber. Dtsch. Math.-Ver., 122 (2020), 35-64.
doi: 10.1365/s13291-019-00210-z. |
[14] |
X. Li,
On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.
doi: 10.1016/j.nonrwa.2019.02.005. |
[15] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[16] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvac, 44 (2001), 441–469. URL http://www.math.kobe-u.ac.jp/ fe/xml/mr1893940.xml. |
[17] |
K. J. Painter,
Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theor. Biol., 481 (2019), 162-182.
doi: 10.1016/j.jtbi.2018.06.019. |
[18] |
R. B. Salako and W. Shen,
Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.
doi: 10.1142/S0218202518400146. |
[19] |
R. B. Salako and W. Shen,
Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. II. Existence, uniqueness, and stability of strictly positive entire solutions, J. Math. Anal. Appl., 464 (2018), 883-910.
doi: 10.1016/j.jmaa.2018.04.034. |
[20] |
R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^n$. III. Transition fronts, Preprint, arXiv: 1811.01525. Google Scholar |
[21] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-368.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[22] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[23] |
G. Viglialoro,
Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.
doi: 10.1016/j.jmaa.2016.02.069. |
[24] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[25] |
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. |
[26] |
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. |
[27] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[28] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Mathématiques Pures Appliquées, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[29] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[30] |
M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Für Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[31] |
M. Winkler,
How strong singularities can be regularized by logistic degradation in the Keller–Segel system?, Ann. Mat. Pura Ed Appl., 198 (2019), 1615-1637.
doi: 10.1007/s10231-019-00834-z. |
[32] |
M. Winkler,
The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in ${L}^1$, Adv. Nonlinear Anal., 9 (2019), 526-566.
doi: 10.1515/anona-2020-0013. |
[33] |
D. Woodward, R. Tyson, M. Myerscough, J. Murray, E. Budrene and H. Berg,
Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189.
doi: 10.1016/S0006-3495(95)80400-5. |
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