@article{AIHPC_2003__20_1_53_0,
author = {Del Pino, Manuel and Kowalczyk, Micha\l\ and Wei, Juncheng},
title = {Multi-bump ground states of the Gierer-Meinhardt system in $\mathbb {R}^2$},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
volume = {20},
year = {2003},
pages = {53-85},
zbl = {01901027},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPC_2003__20_1_53_0}
}
del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng. Multi-bump ground states of the Gierer-Meinhardt system in $\mathbb {R}^2$. Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) pp. 53-85. http://gdmltest.u-ga.fr/item/AIHPC_2003__20_1_53_0/
[1] , , Dynamics of an interior spike in the Gierer-Meinhardt system, SIAM J. Math. Anal. 33 (1) (2001) 172-193. | MR 1858874 | Zbl 0993.35012
[2] M. del Pino, P.L. Felmer, M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Asymptotic Anal., to appear.
[3] , , , The Gierer-Meinhardt system: the breaking of symmetry of homoclinics and multi-bump ground states, Comm. Contemp. Math. 3 (3) (2001) 419-439. | MR 1849649 | Zbl 1003.34025
[4] , , , Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J. 50 (5) (2001) 443-507. | MR 1857043 | Zbl 0994.35058
[5] , , A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972) 30-39.
[6] , , Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998) 443-474. | MR 1658569 | Zbl 0955.35024
[7] , , , On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations 26 (2001) 1929-1946. | MR 1876408 | Zbl 0997.35021
[8] , Multi-peak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996) 739-769. | MR 1408543 | Zbl 0866.35039
[9] C. Gui, C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, to appear. | MR 1900999 | Zbl 01738273
[10] , , Multiple interior peak solutions for some singular perturbation problems, J. Differential Equations 158 (1999) 1-27. | MR 1721719 | Zbl 1061.35502
[11] , , On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000) 522-538. | MR 1758231 | Zbl 0949.35052
[12] , , , Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 47-82. | Numdam | MR 1743431 | Zbl 0944.35020
[13] , , , The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D. 150 (1-2) (2001) 25-62. | MR 1818735 | Zbl 0983.35020
[14] , Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and approximate invariant manifold, Duke Math. J. 98 (1999) 59-111. | MR 1687412 | Zbl 0962.35063
[15] , Activators and inhibitors in pattern formation, Stud. Appl. Math. 59 (1978) 1-23. | MR 479051 | Zbl 0407.92023
[16] , , , Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988) 1-27. | MR 929196 | Zbl 0676.35030
[17] , The Algorithmic Beauty of Sea Shells, Springer, Berlin, 1998. | MR 1325695 | Zbl 1011.00506
[18] , Models of Biological Pattern Formation, Academic Press, London, 1982.
[19] , Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc. 45 (1998) 9-18. | MR 1490535 | Zbl 0917.35047
[20] , , On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc. 297 (1986) 351-368. | MR 849484 | Zbl 0635.35031
[21] , , On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991) 819-851. | MR 1115095 | Zbl 0754.35042
[22] , , Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247-281. | MR 1219814 | Zbl 0796.35056
[23] , , Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math. 12 (1995) 327-365. | MR 1337211 | Zbl 0843.35006
[24] , , , Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992) 1-20. | MR 1174600 | Zbl 0785.35041
[25] W.-M. Ni, I. Takagi, E. Yanagida, Stability analysis of point-condensation solutions to a reaction-diffusion system proposed by Gierer-Meinhardt, Tohoku Math. J., to appear.
[26] , , , Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math. 18 (2) (2001) 259-272. | MR 1842911 | Zbl 01652746
[27] W.-M. Ni, I. Takagi, J. Wei, E. Yanagida, in preparartion.
[28] , Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982) 555-593. | MR 661590 | Zbl 0501.35010
[29] , Point-condensation for a reaction-diffusion system, J. Differential Equations 61 (1986) 208-249. | MR 823402 | Zbl 0627.35049
[30] , The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952) 37-72.
[31] , On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations 134 (1997) 104-133. | MR 1429093 | Zbl 0873.35007
[32] , Uniqueness and eigenvalue estimates of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 1457-1480. | MR 1869645 | Zbl 01731207
[33] , On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999) 353-378. | MR 1713076 | Zbl 1014.35005
[34] , Point-condensations generated by Gierer-Meinhardt system: a brief survey, in: , , , (Eds.), New Trends in Nonlinear Partial Differential Equations, 2000, pp. 46-59.
[35] , , On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal. 30 (1999) 1241-1263. | MR 1718301 | Zbl 0955.35006
[36] , , On multiple spike solutions for the two-dimensional Gierer-Meinhardt system: the strong coupling case, J. Differential Equations 178 (2002) 478-518. | MR 1879835 | Zbl 1042.35005
[37] , , Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Science 6 (2001) 415-458. | MR 1871278 | Zbl 01747952