Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

Previous Article
On the solvability of singular boundary value problems on the real line in the critical growth case
DCDS Home
This Issue
Next Article
Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

February 2020, 40(2): 1107-1130. doi: 10.3934/dcds.2020072

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

Lianzhang Bao ^1,2,3,, and Wenxian Shen ^3,

1.	School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
2.	School of Mathematics, Jilin University, Changchun, Jilin 130012, China
3.	Department of Mathematics and Statistics, Auburn University, AL 36849, USA

^* Corresponding author: Lianzhang Bao

Received May 2019 Revised August 2019 Published November 2019

Fund Project: The first author is supported by China Postdoctoral Science Foundation (183816). The second author is supported by NSF grant DMS–1645673.

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $ \mathbb{R}^+ $, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

Keywords: Chemoattraction-repulsion system, nonlinear parabolic equations, free boundary problem, spreading-vanishing dichotomy, invasive population.

Mathematics Subject Classification: Primary: 35R35, 35J65, 35K20; Secondary: 92B05.

Citation: Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072

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.
[2]	G. Bunting, Y.-H. Du and K. Kratowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.
[3]	J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.
[4]	J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389.
[5]	E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Applied Mathematics Letters, 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007.
[6]	E. Galakhov, O. Salieva and J. I. 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.
[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]	D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165.
[9]	D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.
[10]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[11]	H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, Journal of Mathematical Analysis and Applications, 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[12]	K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017.
[13]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[14]	E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[15]	K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, Journal of Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002.
[16]	J. Liu and Z. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, Journal of Biological Dynamics, 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.
[17]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.
[18]	T. Nagai, T. Senba and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
[19]	R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225. doi: 10.3934/dcds.2017268.
[20]	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, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273. doi: 10.1142/S0218202518400146.
[21]	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.
[22]	R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^N$, Journal of Dynamics and Differential Equations, 31 (2019), 1301-1325. doi: 10.1007/s10884-017-9602-6.
[23]	R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011.
[24]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
[25]	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.
[26]	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.
[27]	Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024.
[28]	Y. Wang and Z.-Y. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031.
[29]	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.
[30]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[31]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[32]	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.
[33]	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.
[34]	T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015, 1125–1133. doi: 10.3934/proc.2015.1125.
[35]	Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, Z.Angew. Math. Mech, 96 (2016), 570-584. doi: 10.1002/zamm.201400311.
[36]	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.
[37]	P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202. doi: 10.1016/j.camwa.2016.08.028.

show all references

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.
[2]	G. Bunting, Y.-H. Du and K. Kratowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.
[3]	J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.
[4]	J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389.
[5]	E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Applied Mathematics Letters, 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007.
[6]	E. Galakhov, O. Salieva and J. I. 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.
[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]	D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165.
[9]	D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.
[10]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[11]	H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, Journal of Mathematical Analysis and Applications, 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[12]	K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017.
[13]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[14]	E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[15]	K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, Journal of Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002.
[16]	J. Liu and Z. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, Journal of Biological Dynamics, 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.
[17]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.
[18]	T. Nagai, T. Senba and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
[19]	R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225. doi: 10.3934/dcds.2017268.
[20]	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, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273. doi: 10.1142/S0218202518400146.
[21]	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.
[22]	R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^N$, Journal of Dynamics and Differential Equations, 31 (2019), 1301-1325. doi: 10.1007/s10884-017-9602-6.
[23]	R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690. doi: 10.1016/j.jde.2017.02.011.
[24]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
[25]	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.
[26]	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.
[27]	Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024.
[28]	Y. Wang and Z.-Y. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031.
[29]	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.
[30]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[31]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[32]	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.
[33]	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.
[34]	T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015, 1125–1133. doi: 10.3934/proc.2015.1125.
[35]	Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, Z.Angew. Math. Mech, 96 (2016), 570-584. doi: 10.1002/zamm.201400311.
[36]	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.
[37]	P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202. doi: 10.1016/j.camwa.2016.08.028.

[1]	Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240
[2]	Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121
[3]	Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317
[4]	Runlin Hu, Pan Zheng. On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022041
[5]	Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154
[6]	Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
[7]	Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920
[8]	Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[9]	Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[10]	Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks and Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583
[11]	H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[12]	Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic and Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667
[13]	Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
[14]	R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[15]	Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174
[16]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[17]	Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[18]	Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629
[19]	Runlin Hu, Pan Zheng, Zhangqin Gao. Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022018
[20]	G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Lianzhang Bao Wenxian Shen
on Google Scholar
Lianzhang Bao Wenxian Shen

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Lianzhang Bao Wenxian Shen
on Google Scholar
Lianzhang Bao Wenxian Shen

[Back to Top]