A priori estimates of stationary solutions of an activator-inhibitor system

Indiana University Mathematics Journal

Volumn 56, Issue 2, 2007, Pages 681-732

  Jiang, Huiqiang ^a   Ni, Wei Ming ^a  

^a University of Minnesota   (United States)

Author keywords

A priori estimate; Activator inhibitor; Existence; Gierer meinhardt; Reaction diffusion

Indexed keywords

EID: 34249819057     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2007.56.2982     Document Type: Article

References (16)

1. MANUELA. DEL VINO, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system, Indiana Univ. Math. J. 43 (1994), 77-129, http://dx.doi.org/10.1512/iumj. 1994.43.43005. MR 1275454 (95d:35050)
2. ALFRED GIERER and HANS MEINHARDT, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39, http://dx.doi.org/10.1007/BF00289234.
3. HUIQIANG JIANG, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst. 14 (2006), 737-751. MR 2177095 (2006g:35107)
4. MING DE LI, SHAG HUA CHEN, and YU CHUN QIN, Boundedness and blow up for the general activator-inhibitor model, Acta Math. Appl. Sinica (English Ser.) 11 (1995), 59-68, http://dx.doi.org/10.1007/BF02012623. MR 1326645 (95m:35189)
5. KYÜYA MASUDA and KATSUO TAKAHASHI, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math. 4 (1987), 47-58. MR 899203 (88j:92032)
6. WEI-MING NI, KANAKO SUZUKI, and IZUMI TAKAGI, The complete dynamics of a kinetic activator-inhibitor system (preprint).
7. WEI-MLNG NI and IZUMI TAKAGI, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc. 297 (1986), 351-368, http://dx.doi.org/10.2307/2000473. MR 849484 (87k:35091)
8. _, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math. 12 (1995), 327-365. MR 1337211 (96g:35098)
9. FRANZ ROTHE, Global Solutions of Reaction-diffusion Systems, Lecture Notes in Mathematics, vol. 1072, Springer-Verlag, Berlin, 1984, ISBN 3-540-13365-8. MR 755878 (86d:35071)
10. IZUMI TAKAGI, A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt, Tôhoku Math. J. (2) 34 (1982), 113-132. MR 651710 (83B5012)
11. _, Point-condensation for a reaction-diffusion system, J. Differential Equations 61 (1986), 208-249, http://dx.doi.org/10.1016/ 0022-0396(86)90119-1. MR 823402 (87e:34031)
12. ABRAHAM TREMBLEY, Memoires pour servir a l'histoire d'un genre de polypes d'eau douce, a bras en forme de cornes (1744).
13. ALAN MATHISON TURING, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B), vol. 237, 1952, pp. 37-72.
14. JUNCHENG WEI and MATTHIAS WINTER, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci. 11 (2001), 415-458. MR 1871278 (2002k:35160)
15. _, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal. 30 (1999), 1241-1263 (electronic), http://dx.doi.org/10.1137/S0036141098347237. MR 1718301 (2000k:92005)
16. _, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations 178 (2002), 478-518, http://dx.doi.org/10.1006/jdeq.2001.4019. MR 1879835 (2002m:35095)