August  2021, 26(8): 4093-4109. doi: 10.3934/dcdsb.2020275

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

* Corresponding author: Mario Fuest

Received  April 2020 Revised  July 2020 Published  August 2021 Early access  September 2020

Fund Project: The first author has been supported by China Scholarship Council (No. 201906090124), and in part by National Natural Science Foundation of China (Nos. 11671079, 11701290, 11601127 and 11171063), and the Natural Science Foundation of Jiangsu Province (No. BK20170896). The second author is partially supported by the German Academic Scholarship Foundation and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140

We construct global generalized solutions to the chemotaxis system
$ \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} $
in smooth, bounded domains
$ \Omega \subset \mathbb R^n $
,
$ n \geq 2 $
, for certain choices of
$ \lambda, \mu $
and
$ \kappa $
.
Here, inter alia, the selections
$ \mu(x) = |x|^\alpha $
with
$ \alpha < 2 $
and
$ \kappa = 2 $
as well as
$ \mu \equiv \mu_1 > 0 $
and
$ \kappa > \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\} $
are admissible (in both cases for any sufficiently smooth
$ \lambda $
).
While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1,526–566), where the condition
$ \kappa > \frac{2n+4}{n+4} $
has been imposed. In particular, for
$ n = 2 $
, our result shows that taking any
$ \kappa > 1 $
suffices to exclude the possibility of collapse into a persistent Dirac distribution.
Citation: Jianlu Yan, Mario Fuest. When do Keller–Segel systems with heterogeneous logistic sources admit generalized solutions?. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4093-4109. doi: 10.3934/dcdsb.2020275
References:
[1]

N. BellomoA. BellouquidY. 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.

[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).

[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. OsakiT. TsujikawaA. 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.

[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. ShigesadaK. 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. WoodwardR. TysonM. MyerscoughJ. MurrayE. 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. BellomoA. BellouquidY. 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.

[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).

[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. OsakiT. TsujikawaA. 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.

[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. ShigesadaK. 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. WoodwardR. TysonM. MyerscoughJ. MurrayE. 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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