# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022121
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## Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation

 School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

* Corresponding author: Wenji Zhang

Received  January 2022 Revised  May 2022 Early access June 2022

Fund Project: The author was supported by the Scientific Research Funds of Hunan Provincial Education Department (No. 21C0356)

This paper considers the Neumann initial-boundary value problem for the chemotaxis system with singular sensitivity
 $\begin{equation*} \begin{split} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi\nabla \cdot (\frac{u}{v}\nabla v) + f(u),}&{x \in \Omega ,t > 0,} \\ {{v_t} = \Delta v - v + u,}&{x \in \Omega ,t > 0,} \end{array}} \right. \end{split} \end{equation*}$
in a smooth bounded domain
 $\Omega \subset {\mathbb{R}^{n}}$
 $(n\geq2)$
, where
 $f\in C^{1}([0,\infty))$
generalizes the logistic function
 $f(s) = \lambda s-\mu s^{\alpha}$
, with
 $\lambda\geq 0$
,
 $\mu>0$
and
 $\alpha>1$
. We prove global existence of solutions to this system in an appropriately generalized sense for any
 $\chi>0$
and
 $\alpha>1$
.
Citation: Wenji Zhang. Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022121
##### References:
 [1] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. [2] X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141. [3] M. Ding, W. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009. [4] M. Fuest, Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening, Nonlinear Differ. Equ. Appl., 28 (2021), Paper No. 16, 17 pp. doi: 10.1007/s00030-021-00677-9. [5] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045. [6] K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81. [7] K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149. [8] K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017. [9] X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058. [10] E. F. Keller and L. A. Segel, Travelling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. [11] E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012. [12] E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0. [13] J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489. [14] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [15] J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 49, 33 pp. doi: 10.1007/s00030-017-0472-8. [16] T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal., 30 (1997), 3837-3842.  doi: 10.1016/S0362-546X(96)00256-8. [17] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156. [18] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [19] K. Osaki and A. Yagi, Global existence of a chemotaxis-growth system in $R^{2}$, Adv. Math. Sci. Appl., 12 (2002), 587-606. [20] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [21] C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006. [22] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [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, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [25] 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. [26] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346. [27] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [28] 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. [29] M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135. [30] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [31] M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$, Adv. Nonlin. Anal., 9 (2020), 526-566.  doi: 10.1515/anona-2020-0013. [32] M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, Preprint. [33] J. Yan and M. Fuest, When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4093-4109.  doi: 10.3934/dcdsb.2020275. [34] X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13 pp. doi: 10.1007/s00033-016-0749-5. [35] X. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.

show all references

##### References:
 [1] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. [2] X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141. [3] M. Ding, W. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009. [4] M. Fuest, Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening, Nonlinear Differ. Equ. Appl., 28 (2021), Paper No. 16, 17 pp. doi: 10.1007/s00030-021-00677-9. [5] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045. [6] K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81. [7] K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149. [8] K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017. [9] X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058. [10] E. F. Keller and L. A. Segel, Travelling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. [11] E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012. [12] E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0. [13] J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489. [14] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [15] J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 49, 33 pp. doi: 10.1007/s00030-017-0472-8. [16] T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal., 30 (1997), 3837-3842.  doi: 10.1016/S0362-546X(96)00256-8. [17] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156. [18] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [19] K. Osaki and A. Yagi, Global existence of a chemotaxis-growth system in $R^{2}$, Adv. Math. Sci. Appl., 12 (2002), 587-606. [20] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [21] C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006. [22] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [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, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [25] 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. [26] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346. [27] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [28] 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. [29] M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135. [30] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [31] M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$, Adv. Nonlin. Anal., 9 (2020), 526-566.  doi: 10.1515/anona-2020-0013. [32] M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, Preprint. [33] J. Yan and M. Fuest, When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 4093-4109.  doi: 10.3934/dcdsb.2020275. [34] X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13 pp. doi: 10.1007/s00033-016-0749-5. [35] X. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.
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