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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

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