April  2014, 7(2): 307-316. doi: 10.3934/dcdss.2014.7.307

Two-dimensional individual clustering model

1. 

Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse cedex 9, France

Received  February 2013 Revised  April 2013 Published  September 2013

This paper is devoted to study a model of individual clustering with two specific reproduction rates in two space dimensions. Given $q>2$ and an initial condition in $W^{1,q}(\Omega)$, the local existence and uniqueness of solution have been shown in [6]. In this paper we give a detailed proof of existence of global solution.
Citation: Elissar Nasreddine. Two-dimensional individual clustering model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 307-316. doi: 10.3934/dcdss.2014.7.307
References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Volume I . Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations," Oxford Lecture Series in Mathematics and it Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.

[3]

J. P. Dias, A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal, J. Math. Anal. Appl., 67 (1979), 525-541. doi: 10.1016/0022-247X(79)90041-6.

[4]

J. P. Dias, Un problème aux limites pour un système d'équations non linéaires tridimensionnel, Bolletino U. M. I. B (5), 16 (1979), 22-31.

[5]

P. Grindrod, Models of individual aggregation or clustering in single and multi-species communities, J. Math. Biol., 26 (1988), 651-660. doi: 10.1007/BF00276146.

[6]

E. Nasreddine, Well-posedness for a model of individual clustering, arXiv:1211.2969v1, (2012).

[7]

M. Schoenauer, Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées, Annales de la Faculté des Sciences de Toulouse 5e Série, 2 (1980), 125-135. doi: 10.5802/afst.550.

[8]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

show all references

References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Volume I . Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations," Oxford Lecture Series in Mathematics and it Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.

[3]

J. P. Dias, A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal, J. Math. Anal. Appl., 67 (1979), 525-541. doi: 10.1016/0022-247X(79)90041-6.

[4]

J. P. Dias, Un problème aux limites pour un système d'équations non linéaires tridimensionnel, Bolletino U. M. I. B (5), 16 (1979), 22-31.

[5]

P. Grindrod, Models of individual aggregation or clustering in single and multi-species communities, J. Math. Biol., 26 (1988), 651-660. doi: 10.1007/BF00276146.

[6]

E. Nasreddine, Well-posedness for a model of individual clustering, arXiv:1211.2969v1, (2012).

[7]

M. Schoenauer, Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées, Annales de la Faculté des Sciences de Toulouse 5e Série, 2 (1980), 125-135. doi: 10.5802/afst.550.

[8]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

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