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doi: 10.3934/eect.2022018
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## Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production

 1 College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

* Corresponding author: Pan Zheng

Received  November 2021 Revised  February 2022 Early access April 2022

This paper deals with a quasilinear parabolic-elliptic chemo-repulsion system with nonlinear signal production
 $\begin{eqnarray*} \label{1a} \left\{ \begin{split}{} & u_t = \nabla\cdot(\phi(u)\nabla u)+\chi\nabla\cdot(u(u+1)^{\alpha-1}\nabla v)+f(u), & (x,t)\in \Omega\times (0,\infty), \\ & 0 = \Delta v-v+u^{\beta}, & (x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a smoothly bounded domain
 $\Omega \subset \mathbb{R}^{n}(n\geq1),$
where
 $\chi,\beta>0,\alpha\in\mathbb{R},$
the nonlinear diffusion
 $\phi\in C^{2}([0,\infty))$
satisfies
 $\phi(u)\geq(u+1)^{m}$
with
 $m\in\mathbb{R},$
and the function
 $f\in C^{1}([0,\infty))$
is a generalized growth term.
 $\bullet$
When
 $f\equiv0,$
it is shown that the solution of the above system is global and uniformly bounded for all
 $\chi,\beta>0$
and
 $m,\alpha\in\mathbb{R}$
.
 $\bullet$
When
 $f\not\equiv0$
and assume that
 $f(u)\leq ku-bu^{\gamma+1}$
with
 $k,b,\gamma>0,$
it is proved that the solution of the above system is also global and uniformly bounded for all
 $\chi,\beta>0$
and
 $m,\alpha\in\mathbb{R}.$
Citation: Runlin Hu, Pan Zheng, Zhangqin Gao. Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production. Evolution Equations and Control Theory, doi: 10.3934/eect.2022018
##### References:
 [1] N. D. Alikakos, $L^{p}$-bounds of solutions of reaction-diffusion equations, Commun. Partial. Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113. [2] T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems, Z. Angew. Math. Phys., 72 (2021), Paper No. 96, 23 pp. doi: 10.1007/s00033-021-01524-8. [3] Y. Chiyo and T. Yokota, Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system, Z. Angew. Math. Phys., 73 (2022), Paper No. 61. doi: 10.1007/s00033-022-01695-y. [4] M. Fuest, Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 16, 17 pp. doi: 10.1007/s00030-021-00677-9. [5] G. L. Hazelbauer, Taxis and behavior: Elementary sensory systems in biology, Chapman and Hall, London, 3 (1979), 185-186. [6] F. Heihoff, On the existence of global smooth solutions to the parabolic-elliptic Keller-Segel system with irregular initial data, J. Dyn. Differ. Equ., 9 (2021), 1-25. [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] L. Hong, M. Tian and S. Zheng, An attraction-repulsion chemotaxis system with nonlinear productions, J. Math. Anal. Appl., 484 (2020), Paper No. 123703, 8 pp. doi: 10.1016/j.jmaa.2019.123703. [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] R. Hu and P. Zheng, On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production, Discrete Contin. Dyn. Syst. Ser. B., 2022. doi: 10.3934/dcdsb.2022041. [11] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis growth system, Nonlinear Anal. Theory Methods Appl., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [12] Y. Lai and Y. Xiao, Existence and asymptotic behavior of global solutions to chemorepulsion systems with nonlinear sensitivity, Electron. J. Differ. Equ., (2017), Paper No. 254, 9 pp. [13] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. B., 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [14] H. Lefraich, L. Taourirte, H. Khalfi and N. E. Alaa, On the existence of global weak solutions to a generalized Keller-Segel model with arbitrary growth and nonlinear signal production, An. Univ. Craiova Ser. Mat. Inform., 46 (2019), 99-108. [15] J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051. [16] Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942. [17] K. Lin and T. Xiang, Strong damping effect of chemo-repulsion prevents blow-up, J. Math. Phys., 62 (2021), Paper No. 041508, 29 pp. doi: 10.1063/5.0032829. [18] M. S. Mock, An intial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612.  doi: 10.1137/0505061. [19] M. S. Mock, Asymtotic behaviour of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225.  doi: 10.1016/0022-247X(75)90172-9. [20] T. Nagai and T. Yamada, Global existence of solutions to the Cauchy problem for an attraction-repulsion chemotaxis system in $\mathbb{R}^{2}$ in the attractive dominant case, J. Math. Anal. Appl., 462 (2018), 1519-1535.  doi: 10.1016/j.jmaa.2018.02.057. [21] Y. Tanaka, Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, J. Math. Anal. Appl., 506 (2022), Paper No. 125654, 29 pp. doi: 10.1016/j.jmaa.2021.125654. [22] Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705. [23] Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [24] Y. Tao and M. Winkler, A chmotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943. [25] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [26] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial. Differ. Equ., 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [27] R. Temam, Infinite-Dimensional Dynamical Systemsin Mechanics and Physics, 2$^nd$ edition, Appl. Math. Sci. vol.68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [28] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [29] W. Wang and Y. Li, Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production, Nonlinear Anal. Real World Appl., 59 (2021), Paper No. 103237, 21 pp. doi: 10.1016/j.nonrwa.2020.103237. [30] W. Wang, M. Zhuang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differ. Equ., 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011. [31] Y. Wang, Boundedness in a quasilinear parabolic-elliptic repulsion chemotaxis system with logistic source, 2014 11th ICCWAMTIP. IEEE., (2014), 373–376. [32] Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Prob., (2016), Paper No. 9, 22 pp. doi: 10.1186/s13661-016-0518-6. [33] 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. [34] 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. [35] 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. [36] 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. [37] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e. [38] 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. [39] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [40] T. Xiang, Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion, Commun. Pure Appl. Anal., 18 (2019), 255-284.  doi: 10.3934/cpaa.2019014. [41] Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311. [42] P. Zheng, On a generalized volume-filling chemotaxis system with nonlinear signal production, Monatsh. Math., 2022. doi: 10.1007/s00605-022-01669-2. [43] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031. [44] S. Zhou, T. Gong and J. Yang, Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension, Appl. Math. J. Chinese Univ. Ser. B., 35 (2020), 244-252.  doi: 10.1007/s11766-020-3994-5.

show all references

##### References:
 [1] N. D. Alikakos, $L^{p}$-bounds of solutions of reaction-diffusion equations, Commun. Partial. Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113. [2] T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems, Z. Angew. Math. Phys., 72 (2021), Paper No. 96, 23 pp. doi: 10.1007/s00033-021-01524-8. [3] Y. Chiyo and T. Yokota, Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system, Z. Angew. Math. Phys., 73 (2022), Paper No. 61. doi: 10.1007/s00033-022-01695-y. [4] M. Fuest, Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 16, 17 pp. doi: 10.1007/s00030-021-00677-9. [5] G. L. Hazelbauer, Taxis and behavior: Elementary sensory systems in biology, Chapman and Hall, London, 3 (1979), 185-186. [6] F. Heihoff, On the existence of global smooth solutions to the parabolic-elliptic Keller-Segel system with irregular initial data, J. Dyn. Differ. Equ., 9 (2021), 1-25. [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] L. Hong, M. Tian and S. Zheng, An attraction-repulsion chemotaxis system with nonlinear productions, J. Math. Anal. Appl., 484 (2020), Paper No. 123703, 8 pp. doi: 10.1016/j.jmaa.2019.123703. [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] R. Hu and P. Zheng, On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production, Discrete Contin. Dyn. Syst. Ser. B., 2022. doi: 10.3934/dcdsb.2022041. [11] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis growth system, Nonlinear Anal. Theory Methods Appl., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [12] Y. Lai and Y. Xiao, Existence and asymptotic behavior of global solutions to chemorepulsion systems with nonlinear sensitivity, Electron. J. Differ. Equ., (2017), Paper No. 254, 9 pp. [13] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. B., 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [14] H. Lefraich, L. Taourirte, H. Khalfi and N. E. Alaa, On the existence of global weak solutions to a generalized Keller-Segel model with arbitrary growth and nonlinear signal production, An. Univ. Craiova Ser. Mat. Inform., 46 (2019), 99-108. [15] J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051. [16] Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942. [17] K. Lin and T. Xiang, Strong damping effect of chemo-repulsion prevents blow-up, J. Math. Phys., 62 (2021), Paper No. 041508, 29 pp. doi: 10.1063/5.0032829. [18] M. S. Mock, An intial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612.  doi: 10.1137/0505061. [19] M. S. Mock, Asymtotic behaviour of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225.  doi: 10.1016/0022-247X(75)90172-9. [20] T. Nagai and T. Yamada, Global existence of solutions to the Cauchy problem for an attraction-repulsion chemotaxis system in $\mathbb{R}^{2}$ in the attractive dominant case, J. Math. Anal. Appl., 462 (2018), 1519-1535.  doi: 10.1016/j.jmaa.2018.02.057. [21] Y. Tanaka, Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, J. Math. Anal. Appl., 506 (2022), Paper No. 125654, 29 pp. doi: 10.1016/j.jmaa.2021.125654. [22] Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705. [23] Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [24] Y. Tao and M. Winkler, A chmotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943. [25] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [26] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial. Differ. Equ., 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [27] R. Temam, Infinite-Dimensional Dynamical Systemsin Mechanics and Physics, 2$^nd$ edition, Appl. Math. Sci. vol.68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [28] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [29] W. Wang and Y. Li, Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production, Nonlinear Anal. Real World Appl., 59 (2021), Paper No. 103237, 21 pp. doi: 10.1016/j.nonrwa.2020.103237. [30] W. Wang, M. Zhuang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differ. Equ., 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011. [31] Y. Wang, Boundedness in a quasilinear parabolic-elliptic repulsion chemotaxis system with logistic source, 2014 11th ICCWAMTIP. IEEE., (2014), 373–376. [32] Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Prob., (2016), Paper No. 9, 22 pp. doi: 10.1186/s13661-016-0518-6. [33] 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. [34] 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. [35] 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. [36] 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. [37] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e. [38] 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. [39] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [40] T. Xiang, Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion, Commun. Pure Appl. Anal., 18 (2019), 255-284.  doi: 10.3934/cpaa.2019014. [41] Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311. [42] P. Zheng, On a generalized volume-filling chemotaxis system with nonlinear signal production, Monatsh. Math., 2022. doi: 10.1007/s00605-022-01669-2. [43] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031. [44] S. Zhou, T. Gong and J. Yang, Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension, Appl. Math. J. Chinese Univ. Ser. B., 35 (2020), 244-252.  doi: 10.1007/s11766-020-3994-5.
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