American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On a comparison method to reaction-diffusion systems and its applications to chemotaxis
  • DCDS-B Home
  • This Issue
  • Next Article
    Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation
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, (2008), 105.  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.  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, 16 (1979), 22.   Google Scholar

[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.  Google Scholar

[6]

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

[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.  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.  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.  doi: 10.1007/BF01762360.  Google Scholar

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, (2008), 105.  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.  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, 16 (1979), 22.   Google Scholar

[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.  Google Scholar

[6]

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

[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.  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.  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.  doi: 10.1007/BF01762360.  Google Scholar

[1]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[2]

Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110
[3]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184
[4]

Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations & Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337
[5]

Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182
[6]

Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[7]

Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
[8]

Dongho Chae. Existence of a semilinear elliptic system with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 709-718. doi: 10.3934/dcds.2007.18.709
[9]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[10]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
[11]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020184
[12]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196
[13]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[14]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[15]

Huiqiang Jiang. Global existence of solutions of an activator-inhibitor system. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 737-751. doi: 10.3934/dcds.2006.14.737
[16]

Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583
[17]

Yuming Qin, Xinguang Yang, Zhiyong Ma. Global existence of solutions for the thermoelastic Bresse system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1395-1406. doi: 10.3934/cpaa.2014.13.1395
[18]

Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258
[19]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[20]

Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences