# American Institute of Mathematical Sciences

January 2019, 18(1): 255-284. doi: 10.3934/cpaa.2019014

## Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion

 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author

Received  November 2017 Revised  March 2018 Published  August 2018

In this work, we are concerned with a class of parabolic-elliptic chemotaxis systems with the prototype given by
 $\left\{ \begin{array}{lll}&u_t = \nabla\cdot(\nabla u-\chi u\nabla v)+au-bu^\theta, &x\in \Omega, t>0, \\&0 = \Delta v -v+u^\kappa, & x\in \Omega, t>0 \end{array}\right.$
with nonnegative initial condition for
 $u$
and homogeneous Neumann boundary conditions in a smooth bounded domain
 $Ω\subset \mathbb{R}^n(n≥ 2)$
, where
 $χ, b, κ>0$
,
 $a∈ \mathbb{R}$
and
 $θ>1$
.
First, using different ideas from [9,11], we re-obtain the boundedness and global existence for the corresponding initial-boundary value problem under, either
 $κ+1<\max\{θ, 1+\frac{2}{n}\}$
or
 $θ = κ+1, \ \ b≥ \frac{(κ n-2)}{κ n}χ.$
Next, carrying out bifurcation from "old multiplicity", we show that the corresponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity
 $χ$
and the emerging patterns converge weakly in
 $L^θ(Ω)$
to some constants as
 $χ \to ∞$
. This provides more details and also fills up a gap left in Kuto et al. [13] for the particular case that
 $θ = 2$
and
 $κ = 1$
. Finally, for
 $θ = κ+1$
, the global stabilities of the equilibria
 $((a/b)^{\frac{1}{κ}}, a/b)$
and
 $(0,0)$
are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small
 $χ$
or large damping rate
 $b$
; on the other hand, they cover and extend He and Zheng's [6,Theorems 1 and 2] for logistic source and linear secretion (
 $θ = 2$
and
 $κ = 1$
) (where convergence rate estimates were shown) to generalized logistic source and nonlinear secretion.
Citation: Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. doi: 10.1512/iumj.2016.65.5776. [2] N. Bellomo, A. Bellouquid, Y. S. 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 and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330. doi: 10.1002/mma.2992. [4] A. Friedman, Partial Differential Equations, Holt, Rinehart Winston, New York, 1969. [5] E. Galakhova, O. Salievab and J. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. [6] 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. [7] T. Hillen and K. 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, From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence Ⅰ, Jahresber DMV, 105 (2003), 103-165. [9] B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7. doi: 10.1016/j.aml.2016.08.003. [10] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966. [11] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. [12] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. [13] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. [14] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. [15] D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388. doi: 10.1007/s11766-016-3386-z. [16] Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559. [17] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. [18] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. [19] Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225. [20] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019. [21] J. Tello and M. Winkler, chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. [22] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. [23] Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204, 2015. [24] M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 38 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071. [25] 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. [26] 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. [27] M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-279. doi: 10.3934/dcdsb.2017135. [28] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032. [29] T. Xiang, On a class of Keller-Segel chemotaxis systems with cross-diffusion, J. Differential Equations, 259 (2015), 4273-4326. doi: 10.1016/j.jde.2015.05.021. [30] J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.

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##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. doi: 10.1512/iumj.2016.65.5776. [2] N. Bellomo, A. Bellouquid, Y. S. 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 and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330. doi: 10.1002/mma.2992. [4] A. Friedman, Partial Differential Equations, Holt, Rinehart Winston, New York, 1969. [5] E. Galakhova, O. Salievab and J. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. [6] 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. [7] T. Hillen and K. 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, From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence Ⅰ, Jahresber DMV, 105 (2003), 103-165. [9] B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7. doi: 10.1016/j.aml.2016.08.003. [10] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966. [11] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. [12] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. [13] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. [14] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. [15] D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388. doi: 10.1007/s11766-016-3386-z. [16] Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559. [17] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. [18] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513. [19] Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225. [20] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161. doi: 10.1016/j.jde.2015.07.019. [21] J. Tello and M. Winkler, chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. [22] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. [23] Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204, 2015. [24] M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 38 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071. [25] 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. [26] 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. [27] M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-279. doi: 10.3934/dcdsb.2017135. [28] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032. [29] T. Xiang, On a class of Keller-Segel chemotaxis systems with cross-diffusion, J. Differential Equations, 259 (2015), 4273-4326. doi: 10.1016/j.jde.2015.05.021. [30] J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
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