Comptes Rendus
Partial Differential Equations
On a class of singular Gierer–Meinhardt systems arising in morphogenesis
[Sur une classe de systèmes singuliers de Gierer–Meinhardt avec applications en morphogenèse]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 163-168.

We study the existence or the nonexistence of classical solutions to a singular Gierer–Meinhardt system with Dirichlet boundary condition. The main feature of our model is that the activator and the inhibitor have different sources given by general nonlinearities. Additional regularity and uniqueness results are established for the one-dimensional case.

On étudie l'existence ou la non-existence des solutions classiques pour une classe de systèmes singuliers de Gierer–Meinhardt avec condition de Dirichlet sur le bord. La caractéristique de notre modèle réside dans la présence de sources différentes pour l'activateur et aussi pour l'inhibiteur. Des propriétés supplémentaires de régularité et d'unicité sont établies en dimension 1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.12.008

Marius Ghergu 1 ; Vicenţiu Rădulescu 2

1 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
2 Department of Mathematics, University of Craiova, RO-200585 Craiova, Romania
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Marius Ghergu; Vicenţiu Rădulescu. On a class of singular Gierer–Meinhardt systems arising in morphogenesis. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 163-168. doi : 10.1016/j.crma.2006.12.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.12.008/

[1] Y.S. Choi; P.J. McKenna A singular Gierer–Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 17 (2000), pp. 503-522

[2] Y.S. Choi; P.J. McKenna A singular Gierer–Meinhardt system of elliptic equations: the classical case, Nonlinear Anal., Volume 55 (2003), pp. 521-541

[3] M. Ghergu; V. Rădulescu On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., Volume 311 (2005), pp. 635-646

[4] M. Ghergu, V. Rădulescu, A singular Gierer–Meinhardt system with different source terms, in preparation

[5] A. Gierer; H. Meinhardt A theory of biological pattern formation, Kybernetik, Volume 12 (1972), pp. 30-39

[6] E.H. Kim A class of singular Gierer–Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., Volume 59 (2004), pp. 305-318

[7] E.H. Kim Singular Gierer–Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., Volume 308 (2005), pp. 1-10

[8] H. Meinhardt; A. Gierer Generation and regeneration of sequence of structures during morphogenesis, J. Theoret. Biol., Volume 85 (1980), pp. 429-450

[9] W.-M. Ni Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., Volume 45 (1998), pp. 9-18

[10] W.-M. Ni; K. Suzuki; I. Takagi The dynamics of a kinetics activato-inhibitor system, J. Differential Equations, Volume 229 (2006), pp. 426-465

[11] W.-M. Ni; J. Wei On positive solutions concentrating on spheres for the Gierer Meinhardt system, J. Differential Equations, Volume 221 (2006), pp. 158-189

[12] A.M. Turing The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B, Volume 237 (1952), pp. 37-72

[13] J. Wei; M. Winter Existence and stability analysis of asymmetric for the Gierer Meinhardt system, J. Math. Pures Appl., Volume 83 (2004), pp. 433-476

[14] J. Wei; M. Winter Spikes for the Gierer–Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations, Volume 178 (2002), pp. 478-518

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