@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}, pages = {53--85}, publisher = {Elsevier}, volume = {20}, number = {1}, year = {2003}, zbl = {01901027}, language = {en}, url = {http://archive.numdam.org/item/AIHPC_2003__20_1_53_0/} }
TY - JOUR AU - del Pino, Manuel AU - Kowalczyk, Michał AU - Wei, Juncheng TI - Multi-bump ground states of the Gierer-Meinhardt system in $\mathbb {R}^2$ JO - Annales de l'I.H.P. Analyse non linéaire PY - 2003 SP - 53 EP - 85 VL - 20 IS - 1 PB - Elsevier UR - http://archive.numdam.org/item/AIHPC_2003__20_1_53_0/ LA - en ID - AIHPC_2003__20_1_53_0 ER -
%0 Journal Article %A del Pino, Manuel %A Kowalczyk, Michał %A Wei, Juncheng %T Multi-bump ground states of the Gierer-Meinhardt system in $\mathbb {R}^2$ %J Annales de l'I.H.P. Analyse non linéaire %D 2003 %P 53-85 %V 20 %N 1 %I Elsevier %U http://archive.numdam.org/item/AIHPC_2003__20_1_53_0/ %G en %F 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) no. 1, pp. 53-85. http://archive.numdam.org/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 | Zbl
, ,[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 | Zbl
, , ,[4] Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J. 50 (5) (2001) 443-507. | MR | Zbl
, , ,[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 | Zbl
, ,[7] On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations 26 (2001) 1929-1946. | MR | Zbl
, , ,[8] Multi-peak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996) 739-769. | MR | Zbl
,[9] C. Gui, C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, to appear. | MR | Zbl
[10] Multiple interior peak solutions for some singular perturbation problems, J. Differential Equations 158 (1999) 1-27. | MR | Zbl
, ,[11] On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000) 522-538. | MR | Zbl
, ,[12] Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 47-82. | Numdam | MR | Zbl
, , ,[13] The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D. 150 (1-2) (2001) 25-62. | MR | Zbl
, , ,[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 | Zbl
,[15] Activators and inhibitors in pattern formation, Stud. Appl. Math. 59 (1978) 1-23. | MR | Zbl
,[16] Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988) 1-27. | MR | Zbl
, , ,[17] The Algorithmic Beauty of Sea Shells, Springer, Berlin, 1998. | MR | Zbl
,[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 | Zbl
,[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 | Zbl
, ,[21] On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991) 819-851. | MR | Zbl
, ,[22] Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247-281. | MR | Zbl
, ,[23] Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math. 12 (1995) 327-365. | MR | Zbl
, ,[24] Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992) 1-20. | MR | Zbl
, , ,[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 | Zbl
, , ,[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 | Zbl
,[29] Point-condensation for a reaction-diffusion system, J. Differential Equations 61 (1986) 208-249. | MR | Zbl
,[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 | Zbl
,[32] Uniqueness and eigenvalue estimates of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 1457-1480. | MR | Zbl
,[33] On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999) 353-378. | MR | Zbl
,[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 | Zbl
, ,[36] On multiple spike solutions for the two-dimensional Gierer-Meinhardt system: the strong coupling case, J. Differential Equations 178 (2002) 478-518. | MR | Zbl
, ,[37] Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Science 6 (2001) 415-458. | MR | Zbl
, ,