On a nonlocal eigenvalue problem

 Access to full text

 Full (PDF)

1. [1] A. Bose - G. Kriegsman, Stability of localized structures in non-local reaction-diffusion equations, Methods Appl. Anal.5 (1998), 351-366. Zbl0935.35064MR1669867
2. [2] E.N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, to appear in: "Proceedings of IMS Workshop on Reaction-Diffusion Systems 2000" . MR1904528
3. [3] M. Del Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system, Indiana Univ. Math. J. 43 (1994), 703-728. Zbl0803.35055MR1291536
4. [4] M. Del Pino - P. Felmer - M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Asymp. Anal., to appear. Zbl1163.35354MR1942277
5. [5] A. Doelman - A. Gardner - T.J. Kaper, Stability of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach, PhysicaD. 122 (1998), 1-36. Zbl0943.34039
6. [6] A. Doelman - A. Gardner - T.J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Technical Report, Center for BioDynamics, Boston University, submitted. Zbl0994.35059
7. [7] A. Doelman - T. Kaper - P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity10 (1997), 523-563. Zbl0905.35044MR1438266
8. [8] P. Freitas, Bifurcation and stability of stationary solutions on nonlocal scalar reaction diffusion equations, J. Dyn. Differential Equations6 (1994), 613-629. Zbl0807.35068MR1303277
9. [9] P. Freitas, A non-local Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburg Sect. A 124 (1994), 169-188. Zbl0798.34033MR1272438
10. [10] P. Freitas, Stability of stationary solutions for a scalar nonlocal reaction diffusion equation, Quart. J. Mech. Appl. Math.48 (1995), 557-582. Zbl0842.35043MR1387092
11. [11] A. Gierer - H. Meinhardt, A theory of biological pattern formation, Kybemetik (Berlin) 12 (1972), 30-39. 
12. [12] P. Gray - 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. 
13. [13] P. Gray - S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B → 3B, B → C, Chem. Eng. Sci.39 (1984), 1087-1097. 
14. [14] J.K. Hale - L.A. Peletier - W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math.61 (2000), 102-130. Zbl0965.34037MR1776389
15. [15] D. Iron - M.J. Ward, A metastable spike solution for a non-local reaction-diffusion model, SIAM J. Appl. Math.60 (2000), 778-802. Zbl0956.35011MR1740850
16. [16] D. Iron - M.J. Ward - J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, PhysicaD., to appear. Zbl0983.35020MR1818735
17. [17] C.-S. Lin - W.-M. Ni, "On the diffusion coefficient of a semilinear Neumann problem", Calculus of variations and partial differential equations (Trento, 1986), 160-174, Lecture Notes in Math., 1340, Springer, Berlin-New York, 1988. Zbl0704.35050MR974610
18. [18] C.B. Muratov - V.V. Osipov, Spike autosolitions in Gray-Scott model, J. Phys. A-Math. Gen.48 (2000), 8893-8916. MR1801475
19. [19] W.-M. Ni - I. Takagi, Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Industrial Appl. Math.12 (1995), 327-365. Zbl0843.35006MR1337211
20. [20] W.-M. Ni - I. Takagi - E. Yanagida, Tohoku Math. J., to appear. 
21. [21] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc.45 (1998), 9-18. Zbl0917.35047MR1490535
22. [22] Y. Nishiura - D. Ueyama, A skeleton structure of self-replicating dynamics, PhysicaD. 130 (1999), 73-104. Zbl0936.35090
23. [23] Y. Nishiura, Global structure of bifurcation solutions of some reaction-diffusion systems, SIAM J. Math. Anal.13 (1982), 555-593. Zbl0501.35010MR661590
24. [24] J. Wei, On the construction ofsingle-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations129 (1996), 315-333. Zbl0865.35011MR1404386
25. [25] J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems, Int. J. Bifur. and Chaos10 (2000), 1485-1496. Zbl1090.35538MR1779676
26. [26] J. Wei - M. Winter, On the two dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal.30 (1999), 1241-1263. Zbl0955.35006MR1718301
27. [27] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness, spectrum estimates and stability analysis, Europ. J. Appl. Math.10 (1999), 353-378. Zbl1014.35005MR1713076
28. [28] J. Wei, Existence, stability and metastability of point condensation patterns generated by Gray-Scott system, Nonlinearity3 (1999), 593-616. Zbl0984.35160MR1690196
29. [29] J. Wei, Point condensations generated by Gierer-Meinhardt system: a brief survey, book chapter, In: "New Trends in Nonlinear Partial Differential Equations 2000", Y. Morita - H. Ninomiya - E. Yanagida - S. Yotsutani (eds), pp. 46-59. 
30. [30] L. Zhang, Uniqueness of positive solutions of Δu + λu + uP = 0 in a ball, Comm. Partial Differential Equations17 (1992), 1141-1164. Zbl0782.35025