American Institute of Mathematical Sciences

Advanced
December  2013, 18(10): 2669-2688. doi: 10.3934/dcdsb.2013.18.2669

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

Mihaela Negreanu 1, and  J. Ignacio Tello 2,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2. 

Departamento de Matemática Aplicada, ETSI SI, Universidad Politécnica de Madrid, 28031 Madrid, Spain

Received  November 2012 Revised  April 2013 Published  October 2013

In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.
Keywords: asymptotic behavior, sub- and super-solutions., Comparison method, stability.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35B5.
Citation: Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669
References:
[1]

S. Ahmad, Convergence and ultimate bound of solutions of the nonautonomous Volterra-Lotka Competition Equations, J. Math. Anal. and Appl., 127 (1987), 377-387. doi: 10.1016/0022-247X(87)90116-8.  Google Scholar

[2]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. American Math. Society, 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.  Google Scholar

[3]

S. Ahmad, Extintion of species in nonautonomous Volterra-Lotka systems, Proc. American Math. Society, 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.  Google Scholar

[4]

S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Analysis, 13 (1998), 263-284. doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[5]

M. Braun, Differential Equations and Their Applications, An introduction to applied mathematics. Fourth edition. Texts in Applied Mathematics, 11. Springer-Verlag, New York, 1993.  Google Scholar

[6]

A. Derlet and P. Takáč, A quasilinear parabolic model for population evolution, Differential equations and Applications, 4 (2012), 121-136. doi: 10.7153/dea-04-08.  Google Scholar

[7]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[8]

M. Negreanu and J. I. Tello, On a Parabolic-Elliptic chemotactic system with non-constant chemotactic sensitivity, Nonlinear Analysis: Theory, Methods & Applications, 80 (2013), 1-13. doi: 10.1016/j.na.2012.12.004.  Google Scholar

[9]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1086-1103. doi: 10.1088/0951-7715/26/4/1083.  Google Scholar

[10]

C. V. Pao, Comparison methods and stability analysis of reaction-diffusion systems, In the book Comparison Methods and Stability Theory, Lecture Notes in Pure and Appl. Math., 162, pp 277-292. Dekker, New York, 1994.  Google Scholar

[11]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Verlag, Basel, 2007.  Google Scholar

[12]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900.  Google Scholar

[13]

P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Equat., 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.  Google Scholar

[14]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two species chemotaxis model, J. Math. Biology, (2013). doi: 10.1007/s00285-013-0681-7.  Google Scholar

[15]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar

[16]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[17]

S. Zheng and H. Su, A quasilinear reaction-diffusion system coupled via nonlocal sources, Applied Mathematics and Computation, 180 (2006), 295-308. doi: 10.1016/j.amc.2005.12.020.  Google Scholar

show all references

References:
[1]

S. Ahmad, Convergence and ultimate bound of solutions of the nonautonomous Volterra-Lotka Competition Equations, J. Math. Anal. and Appl., 127 (1987), 377-387. doi: 10.1016/0022-247X(87)90116-8.  Google Scholar

[2]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. American Math. Society, 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.  Google Scholar

[3]

S. Ahmad, Extintion of species in nonautonomous Volterra-Lotka systems, Proc. American Math. Society, 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.  Google Scholar

[4]

S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Analysis, 13 (1998), 263-284. doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[5]

M. Braun, Differential Equations and Their Applications, An introduction to applied mathematics. Fourth edition. Texts in Applied Mathematics, 11. Springer-Verlag, New York, 1993.  Google Scholar

[6]

A. Derlet and P. Takáč, A quasilinear parabolic model for population evolution, Differential equations and Applications, 4 (2012), 121-136. doi: 10.7153/dea-04-08.  Google Scholar

[7]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[8]

M. Negreanu and J. I. Tello, On a Parabolic-Elliptic chemotactic system with non-constant chemotactic sensitivity, Nonlinear Analysis: Theory, Methods & Applications, 80 (2013), 1-13. doi: 10.1016/j.na.2012.12.004.  Google Scholar

[9]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1086-1103. doi: 10.1088/0951-7715/26/4/1083.  Google Scholar

[10]

C. V. Pao, Comparison methods and stability analysis of reaction-diffusion systems, In the book Comparison Methods and Stability Theory, Lecture Notes in Pure and Appl. Math., 162, pp 277-292. Dekker, New York, 1994.  Google Scholar

[11]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Verlag, Basel, 2007.  Google Scholar

[12]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900.  Google Scholar

[13]

P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Equat., 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.  Google Scholar

[14]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two species chemotaxis model, J. Math. Biology, (2013). doi: 10.1007/s00285-013-0681-7.  Google Scholar

[15]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar

[16]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[17]

S. Zheng and H. Su, A quasilinear reaction-diffusion system coupled via nonlocal sources, Applied Mathematics and Computation, 180 (2006), 295-308. doi: 10.1016/j.amc.2005.12.020.  Google Scholar

[1]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[2]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[3]

Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[4]

Yong Liu. Even solutions of the Toda system with prescribed asymptotic behavior. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1779-1790. doi: 10.3934/cpaa.2011.10.1779
[5]

Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251
[6]

Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[7]

Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463
[8]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
[9]

Yongqin Liu, Shuichi Kawashima. Asymptotic behavior of solutions to a model system of a radiating gas. Communications on Pure & Applied Analysis, 2011, 10 (1) : 209-223. doi: 10.3934/cpaa.2011.10.209
[10]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237
[11]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
[12]

Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253
[13]

Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
[14]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[15]

Georges Chamoun, Moustafa Ibrahim, Mazen Saad, Raafat Talhouk. Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4165-4188. doi: 10.3934/dcdsb.2020092
[16]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041
[17]

Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029
[18]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146
[19]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[20]

Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences