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On a comparison method to reaction-diffusion systems and its applications to chemotaxis
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.
doi: 10.1137/S0036141097318900. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.
doi: 10.1137/S0036141097318900. |
[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. |
[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. |
[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. |
[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. |
[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. |
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