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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

Elissar Nasreddine 1,

1. 

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

Received  November 2012 Revised  February 2013 Published  October 2013

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. Google Scholar

[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, Part 1, 105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008. doi: 10.4064/bc81-0-7.  Google Scholar

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

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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.  Google Scholar

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O. A. Ladyzenskaja, V . A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Providence (R. I.) , American Mathematical Society, 1988. Google Scholar

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M. Langlais and D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal., 9 (1985), 321-333. doi: 10.1016/0362-546X(85)90057-4.  Google Scholar

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M. Schoenauer, Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées, (French) Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 125-135. doi: 10.5802/afst.550.  Google Scholar

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show all references

References:
[1]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and it Applications, 2006. Google Scholar

[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, Part 1, 105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008. doi: 10.4064/bc81-0-7.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

O. A. Ladyzenskaja, V . A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Providence (R. I.) , American Mathematical Society, 1988. Google Scholar

[7]

M. Langlais and D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal., 9 (1985), 321-333. doi: 10.1016/0362-546X(85)90057-4.  Google Scholar

[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), 125-135. doi: 10.5802/afst.550.  Google Scholar

[9]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Annali di Mathematica Pura ed Applicata (IV), 146 (1987), 65-69. doi: 10.1007/BF01762360.  Google Scholar

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