# American Institute of Mathematical Sciences

September  2017, 37(9): 5021-5036. doi: 10.3934/dcds.2017216

## Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions

 Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

* Corresponding author. QW acknowledges the support from National Natural Science Foundation of China (Grant No. 11501460). All authors thank the two anonymous referees for their helpful suggestions, which improve the presentation of this paper

Currently at Department of Mathematics, University of Central Florida, Orlando, USA. yfeng@knights.ucf.edu.

Received  June 2016 Revised  April 2017 Published  June 2017

We consider the following fully parabolic Keller-Segel system
 $\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\v_t=Δ v-v+u,&x ∈ Ω,t>0, \\\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,&x∈\partial Ω,t>0\end{array}\right.$
over a multi-dimensional bounded domain
 $Ω \subset \mathbb R^N$
,
 $N≥2$
. Here
 $D(u)$
and
 $S(u)$
are smooth functions satisfying:
 $D(0)>0$
,
 $D(u)≥ K_1u^{m_1}$
and
 $S(u)≤ K_2u^{m_2}$
,
 $\forall u≥0$
, for some constants
 $K_i∈\mathbb R^+$
,
 $m_i∈\mathbb R$
,
 $i=1, 2$
. It is proved that, when the parameter pair
 $(m_1, m_2)$
lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22,28], in particular when
 $N≥3$
and
 $γ≥1$
, and [3,29] when
 $m_1>γ-\frac{2}{N}$
if
 $γ∈(0, 1)$
or
 $m_1>γ-\frac{4}{N+2}$
if
 $γ∈[1, ∞)$
. Moreover, according to our results, the index
 $\frac{2}{N}$
is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.
Citation: Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216
##### References:
 [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1. [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] X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061. [4] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [5] T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [6] 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. [7] D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences Ⅰ, Jahresber DMV, 105 (2003), 103-165. [8] D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences Ⅱ, Jahresber DMV, 106 (2004), 51-69. [9] 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. [10] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569. [11] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data, J. Differential Equations, 252 (2012), 2469-2491.  doi: 10.1016/j.jde.2011.08.047. [12] E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [13] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [14] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [15] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [16] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. [17] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [18] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [19] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003. [20] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821. [21] 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. [22] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789. [23] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319. [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, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146. [26] 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. [27] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [28] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z. [29] J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with a logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.  doi: 10.1016/j.jmaa.2015.05.071.

show all references

##### References:
 [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1. [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] X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061. [4] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [5] T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [6] 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. [7] D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences Ⅰ, Jahresber DMV, 105 (2003), 103-165. [8] D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences Ⅱ, Jahresber DMV, 106 (2004), 51-69. [9] 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. [10] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569. [11] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data, J. Differential Equations, 252 (2012), 2469-2491.  doi: 10.1016/j.jde.2011.08.047. [12] E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [13] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [14] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [15] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [16] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. [17] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [18] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [19] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003. [20] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821. [21] 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. [22] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789. [23] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319. [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, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146. [26] 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. [27] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [28] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z. [29] J. Zheng, Boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with a logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.  doi: 10.1016/j.jmaa.2015.05.071.
An illustration of parameter regions for the global existence of (1.1) in the $m_1$-$m_2$ plane as $\gamma$ varies. For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slopes of $L_1(L^*_1)$ and $L_2(L^*_2)$ are 1 and the slope of $L_3(L^*_3)$ is $\frac{1}{2}$. Note that in the lower right plot, we use $\frac{1}{N}+\frac{2}{N+2}$, instead of $\frac{3N+2}{N(N+2)}$, for better illustration
An illustration of up-to-date summary results on global existence of the nonlinear diffusion system (1.1). For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slope of $L_1(L^*_1)$ is 1 and of $L^*_2$ is $\frac{1}{2}$; $L_4$ is the horizontal line $m_2=\gamma$
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