Well-posedness for a model of individual clustering

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December 2013, 18(10): 2647-2668. doi: 10.3934/dcdsb.2013.18.2647

Well-posedness for a model of individual clustering

1.	Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse cedex 9

Received November 2012 Revised February 2013 Published October 2013

Abstract
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We study the well-posedness of a model of individual clustering. Given $p>N\geq 1$ and an initial condition in $W^{1,p}(\Omega)$, the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if $N=1$, as well as, the convergence to steady states for one of these rates.

Keywords: compactness method., steady state, global existence, Elliptic system, local existence.

Mathematics Subject Classification: 35K20, 35B40, 35M33, 35J47, 35Q9.

Citation: Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647

References:

[1]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and it Applications, (2006).
[2]	T. Cieślak, Ph. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system,, Parabolic and Navier-Stokes Equations, (2008), 105. doi: 10.4064/bc81-0-7.
[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. 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, 16 (1979), 22.
[5]	P. Grindrod, Models of individual aggregation or clustering in single and multi-species communities,, J. Math. Biol., 26 (1988), 651. doi: 10.1007/BF00276146.
[6]	O. A. Ladyzenskaja, V . A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type,, Providence (R. I.), (1988).
[7]	M. Langlais and D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations,, Nonlinear Anal., 9 (1985), 321. doi: 10.1016/0362-546X(85)90057-4.
[8]	M. Schoenauer, Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées,, (French) Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 125. doi: 10.5802/afst.550.
[9]	J. Simon, Compact sets in the space $L^p(0,T; B)$,, Annali di Mathematica Pura ed Applicata (IV), 146* (1987), 65. doi: 10.1007/BF01762360.*

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References:

[1]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and it Applications, (2006).
[2]	T. Cieślak, Ph. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system,, Parabolic and Navier-Stokes Equations, (2008), 105. doi: 10.4064/bc81-0-7.
[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. 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, 16 (1979), 22.
[5]	P. Grindrod, Models of individual aggregation or clustering in single and multi-species communities,, J. Math. Biol., 26 (1988), 651. doi: 10.1007/BF00276146.
[6]	O. A. Ladyzenskaja, V . A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type,, Providence (R. I.), (1988).
[7]	M. Langlais and D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations,, Nonlinear Anal., 9 (1985), 321. doi: 10.1016/0362-546X(85)90057-4.
[8]	M. Schoenauer, Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées,, (French) Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 125. doi: 10.5802/afst.550.
[9]	J. Simon, Compact sets in the space $L^p(0,T; B)$,, Annali di Mathematica Pura ed Applicata (IV), 146* (1987), 65. doi: 10.1007/BF01762360.*

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