On a class of singular Gierer–Meinhardt systems arising in morphogenesis

Abstract 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. To cite this article: M. Ghergu, V. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

[1]  H. Bhadeshia.  Diffusion , 1995, Theory of Transformations in Steels.

[2]  Matthias Winter,et al.  Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case , 2002 .

[3]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[4]  A class of singular Gierer–Meinhardt systems of elliptic boundary value problems , 2004 .

[5]  Wei-Ming Ni,et al.  DIFFUSION, CROSS-DIFFUSION, AND THEIR SPIKE-LAYER STEADY STATES , 1998 .

[6]  P. J. McKenna,et al.  A singular Gierer-Meinhardt system of elliptic equations: the classical case , 2003 .

[7]  W. Ni,et al.  The dynamics of a kinetic activator–inhibitor system , 2006 .

[8]  Matthias Winter,et al.  Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system , 2004 .

[9]  Singular Gierer–Meinhardt systems of elliptic boundary value problems , 2005 .

[10]  Marius Ghergu,et al.  A singular Gierer—Meinhardt system with different source terms* , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  A Gierer,et al.  Generation and regeneration of sequence of structures during morphogenesis. , 1980, Journal of theoretical biology.

[12]  W. Ni,et al.  On positive solutions concentrating on spheres for the Gierer–Meinhardt system , 2006 .

[13]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[14]  Vicentiu D. Rădulescu,et al.  On a class of sublinear singular elliptic problems with convection term , 2005, math/0504593.

[15]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.