Stable dynamics of spikes in solutions to a system of reaction-diffusion equations

Asymptotic Analysis

Volumn 26, Issue 3-4, 2001, Pages 307-357

  Bellettini, Giovanni ^a   Fusco, Giorgio ^b  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

^b UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035369389     PISSN: 09217134     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (67)

1. Adimurthy, G. Mancini and S.L. Yadava, The role of the mean curvature in semilinear Neumann problems involving critical exponent, Comm. Partial Differential Equations 20 (1995), 591-631.
2. Adimurthy, F. Pacella and S.L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal. 113 (1993), 318-350.
3. ∞-estimates for solutions of a semilinear Neumann problem involving the critical exponent, Differential Integral Equations 8 (1995), 41-68.
4. N.D, Alikakos, L. Bronsard and G. Fusco, Slow motion in the gradient theory of phase transitions via energy and spectrum, Calc. Var. Partial Differential Equations 6 (1998), 39-66.
5. N.D. Alikakos, X. Chen and G. Fusco, Motion of a drop by surface tension along the boundary, Calc. Var. Partial Differential Equations 11(3) (2000), 233-305.
6. N.D. Alikakos and G. Fusco, The spectrum of the Cahn-Hilliard for generic interface in higher space dimensions, Indiana Univ. Math. J. 42 (1998), 637-674.
7. N.D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimension, part II. The motion of bubbles, Arch. Rational Mech. Anal 141 (1998), 1-61.
8. N.D. Alikakos, G. Fusco and M. Kowalczyk, Finite dimensional dynamics and interfaces intersecting the boundary: equilibria and quasi-invariant manifold, Indiana Univ. Math. J. 45 (1996), 1119-1155.
9. N.D. Alikakos, G. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, J. Differential Equations 126 (1996), 106-167.
10. P.W. Bates, N.B. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher dimension and instability, Adv. Diff. Eqns. 4 (1999), 1-69.
11. P.W. Bates and P.C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math. 53 (1993), 990-1008.
12. P.W. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160 (2000), 283-356.
13. P.W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach spaces, Mem. Amer. Math. Soc. 645 (1998).
14. G. Bellettini and G. Fusco, Stable dynamics of spikes in solutions to a system of activator-inhibitor type, in: Proc. of the Workshop "New Trends in Nonlinear Partial Differential Equations", Y. Morita, H. Ninomiya, E. Yanagida and S. Yotsutani, eds, Ryukoku Extension Center, 1999, pp. 1-11.
15. G. Bellettini and G. Fusco, Stable dynamics of spikes in solutions to a system of reaction-diffusion equations II, Paper in preparation.
16. D. Cao, N. Dancer, E. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Cont. Dynam. Systems 2 (1996), 221-236.
17. cursive Greek chicursive Greek chi + f(u), Comm. Pure Appl. Math. 42 (1999), 523-576.
18. X. Chen and M. Kowalczyk, Slow dynamics of internal spikes in the shadow of Gierer-Mainhardt system, Preprint.
19. M. Del Pino, X. Chen and M. Kowalczyk, The Gierer-Meinhardt system: the breaking of homoclinic and multi-bump ground states, Preprint.
20. M. Del Pino, P. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Preprint CMA, 1999.
21. A. Doelman, R.A. Gardner and T.J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Preprint.
22. S.I. Ei, The notion of weakly interacting pulses in reaction-diffusion systems, Preprint.
23. P.C. Fife, On modelling pattern formation by activitor-inhibitor systems, J. Math. Biol. 4 (1977), 353-362.
24. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
25. cursive Greek chicursive Greek chi + f(u) for small ε > 0, in: Lecture Notes in Phys., Vol. 359, 1990, pp. 53-73.
26. G. Fusco and J. Hale, Slow motion manifold, dormant instability, and singular perturbations, J. Dynamics Differential Equations 1 (1989), 75-94.
27. n, in: Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, pp. 369-402.
28. A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39.
29. P. Gray and S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), 29-43.
30. M. Grossi, A. Pistoia and J. Wei, Existence of multiple-peaked solutions in a semilinear elliptic Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11(2) (2000), 143-175.
31. C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1995), 739-769.
32. C. Gui and N. Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical exponent Math. Z. 229 (1998), 443-474.
33. C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbation Neumann problems, J. Differential Equations 158 (1999), 1-27.
34. C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 47-82.
35. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer, 1981.
36. D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meihnardt model, Preprint.
37. E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.
38. M. Kowalczyk, Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and approximate invariant manifold, Duke Math. J. 98 (1999), 59-111.
39. Y.Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations 23(3-4) (1998), 487-545.
40. C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemo-taxis system, J. Differential Equations 72 (1988), 1-27.
41. H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, 1982.
42. W.-M. Ni, Diffusion, cross diffusion, and spike layer solutions, in: Notices Amer. Math. Soc., 1998, pp. 8-18.
43. W.-M. Ni, X. Pan and I. Takagi, Singular behaviour of least-energy solutions to a semilinear Neumann problem involving critical exponent, Duke Math. J. 67 (1992), 1-20.
44. W.-M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 41(1991), 815-819.
45. W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281.
46. W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Ind. Appl. Math. 12 (1995), 327-365.
47. W.-M. Ni, I. Takagi and E. Yanagida, Stability analysis of point-condensation solutions to a reaction-diffusion system proposed by Gierer-Meinhardt, Preprint.
48. W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731-768.
49. Y. Nishiura and D. Ueyama, Physica D 130 (1999), 73-104.
50. X. Pan, Condensation of least energy solutions of a semilinear Neumann problem, J. Partial Differential Equations 8 (1995), 1-35.
51. V. Petrov, S.K. Scott and K. Showalter, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system, Philos. Trans. Roy. Soc. London Ser. A 1685 (1994), 631-642.
52. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, 1975.
53. B. Sandstede, Stability of multi-pulse solutions, Trans. Amer. Math. Soc. 350 (1998), 429-472.
54. B. Sandstede, Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the Fitz-Hugh-Nagumo equation, SIAM J. Math. Anal. 29 (1998), 183-207.
55. A. Turing, The chemical basis of morphogenesism, Philos. Trans. Roy. Soc. (B) 237 (1952), 37-72.
56. X.J. Wang, Neumann problems of semilinear elliptic equations involving critical exponent, J. Differential Equations 93 (1991), 283-310.
57. Z.Q. Wang, On the existence of multiple single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal. 120 (1992), 375-399.
58. Z.Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh Sec. A 125 (1995), 1003-1029.
59. Z.Q. Wang, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear Anal. 127 (1996), 1281-1306.
60. M.J. Ward, An asymptotic analysis of localized solutions for some reaction-diffusion models in multi-dimensional domains, Studies Appl. Math. 97 (1996), 103-126.
61. J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), 315-333.
62. J-Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 131 (1997), 104-133.
63. J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999), 353-378.
64. J. Wei and M. Winter, Multi-boundary-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks, Preprint.
65. J. Wei and M. Winter, On multiple spike solutions for the two-dimensional Gierer-Meinhardt system: the strong coupling case, Preprint.
66. J. Wei and M. Winter, On multiple spike solutions for the two-dimensional Gierer-Meinhardt system: the weak coupling case, Preprint.
67. J. Wei and M. Winter, Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks, Calc. Vor. Partial Differential Equations 10 (2000), 249-289.