# American Institute of Mathematical Sciences

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

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.
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] J. Math. Anal. and Appl., 127 (1987), 377-387. doi: 10.1016/0022-247X(87)90116-8.  Google Scholar [2] Proc. American Math. Society, 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.  Google Scholar [3] Proc. American Math. Society, 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.  Google Scholar [4] Nonlinear Analysis, 13 (1998), 263-284. doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar [5] An introduction to applied mathematics. Fourth edition. Texts in Applied Mathematics, 11. Springer-Verlag, New York, 1993.  Google Scholar [6] Differential equations and Applications, 4 (2012), 121-136. doi: 10.7153/dea-04-08.  Google Scholar [7] J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar [8] Nonlinear Analysis: Theory, Methods & Applications, 80 (2013), 1-13. doi: 10.1016/j.na.2012.12.004.  Google Scholar [9] Nonlinearity, 26 (2013), 1086-1103. doi: 10.1088/0951-7715/26/4/1083.  Google Scholar [10] 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] Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Verlag, Basel, 2007.  Google Scholar [12] SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900.  Google Scholar [13] J. Diff. Equat., 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.  Google Scholar [14] J. Math. Biology, (2013). doi: 10.1007/s00285-013-0681-7.  Google Scholar [15] Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar [16] Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [17] Applied Mathematics and Computation, 180 (2006), 295-308. doi: 10.1016/j.amc.2005.12.020.  Google Scholar

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##### References:
 [1] J. Math. Anal. and Appl., 127 (1987), 377-387. doi: 10.1016/0022-247X(87)90116-8.  Google Scholar [2] Proc. American Math. Society, 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.  Google Scholar [3] Proc. American Math. Society, 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.  Google Scholar [4] Nonlinear Analysis, 13 (1998), 263-284. doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar [5] An introduction to applied mathematics. Fourth edition. Texts in Applied Mathematics, 11. Springer-Verlag, New York, 1993.  Google Scholar [6] Differential equations and Applications, 4 (2012), 121-136. doi: 10.7153/dea-04-08.  Google Scholar [7] J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar [8] Nonlinear Analysis: Theory, Methods & Applications, 80 (2013), 1-13. doi: 10.1016/j.na.2012.12.004.  Google Scholar [9] Nonlinearity, 26 (2013), 1086-1103. doi: 10.1088/0951-7715/26/4/1083.  Google Scholar [10] 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] Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks] Birkhäuser Verlag, Basel, 2007.  Google Scholar [12] SIAM J. Math. Anal., 29 (1998), 1301-1334. doi: 10.1137/S0036141097318900.  Google Scholar [13] J. Diff. Equat., 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.  Google Scholar [14] J. Math. Biology, (2013). doi: 10.1007/s00285-013-0681-7.  Google Scholar [15] Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar [16] Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [17] Applied Mathematics and Computation, 180 (2006), 295-308. doi: 10.1016/j.amc.2005.12.020.  Google Scholar
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