# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020275
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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