Geodesic
Diffusion and cross-diffusion in pattern formation
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 197-214.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We discuss the stability and instability properties of steady state solutions to single equations, shadow systems, as well as $2 \times 2$ systems. Our basic observation is that the more complicated the pattern are, the more unstable they tend to be. 
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Ni, Wei-Ming. Diffusion and cross-diffusion in pattern formation. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 197-214. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a3/

[1] H. Amann, Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems. Diff. Integral Eqns., 3, 1990, 13-75. | MR | Zbl

[2] H. Amann, Nonhomogeneous linear quasilinear elliptic and parabolic boundary value problems. In: H. SCHMEISSER - H. TRIEBEL (eds.), Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Math., 133, Stuttgart-Leipzig 1993, 9-126. | MR | Zbl

[3] H. Amann, Supersolution, monotone iteration and stability. J. Diff. Eqns., 21, 1976, 365-377. | MR | Zbl

[4] R.S. Cantrell - C. Cosner, Spatial ecology via reaction-diffusion equations. John Wiley and Sons, 2003. | DOI | MR | Zbl

[5] R.G. Casten - C.J. Holland, Instability results for a reaction-diffusion equation with Neumann boundary conditions. J. Diff. Eqns., 27, 1978, 266-273. | DOI | MR | Zbl

[6] Y.-S. Choi - R. Lui - Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete and Continuous Dynamical Systems, 10, 2004, 719-730. | DOI | MR | Zbl

[7] A. Gierer - H. Meinhardt, A theory of biological pattern formation. Kybernetik, 12, 1972, 30-39.

[8] J.K. Hale - K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations. Appl. Anal., 32, 1989, 287-303. | DOI | MR | Zbl

[9] J.K. Hale - J.M. Vegas, A nonlinear parabolic equation with varying domain. Arch. Rat. Mech. Anal., 86, 1984, 99-123. | DOI | MR | Zbl

[10] G.E. Hutchinson, An introduction to population ecology. Yale University Press, New Haven, CT 1978. | MR | Zbl

[11] S. Jimbo - Y. Morita, Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels. Comm. PDE, 17, 1992, 523-552. | DOI | MR | Zbl

[12] D. Le, Cross diffusion systems on $n$ spatial dimensional domains. Indiana Univ. Math. J., 51, 2002, 625-643. | DOI | MR | Zbl

[13] C.-S. Lin - W.-M. Ni, Stability of solutions of semilinear diffusion equations. Preprint, 1986.

[14] Y. Lou - W.-M. Ni, Diffusion, self-diffusion and cross-diffusion. J. Diff. Eqns., 131, 1996, 79-131. | DOI | MR | Zbl

[15] Y. Lou - W.-M. Ni, Diffusion vs cross diffusion: An elliptic approach. J. Diff. Eqns., 154, 1999, 157-190. | DOI | MR | Zbl

[16] Y. Lou - W.-M. Ni - Y. Wu, On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 4, 1998, 193-203. | DOI | MR | Zbl

[17] Y. Lou - W.-M. Ni - S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 10, 2004, 435-458. | DOI | MR | Zbl

[18] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS, 15, 1979, 401-454. | fulltext mini-dml | DOI | MR | Zbl

[19] W.-M. Ni, Some aspects of semilinear ellitpic equations. Lecture Notes, National Tsinghua Univ., Hsinchu-Taiwan-China 1987. | Zbl

[20] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states. Notices of Amer. Math. Soc., 45, 1998, 9-18. | MR | Zbl

[21] W.-M. Ni - P. Polacik - E. Yanagida, Montonicity of stable solutions in shadow systems. Trans. Amer. Math. Soc., 353, 2001, 5057-5069. | DOI | MR | Zbl

[22] W.-M. Ni - I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type. Trans. Amer. Math. Soc., 297, 1986, 351-368. | DOI | MR | Zbl

[23] W.-M. Ni - I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math., 44, 1991, 819-851. | DOI | MR | Zbl

[24] W.-M. Ni - I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J., 70, 1993, 247-281. | fulltext mini-dml | DOI | MR | Zbl

[25] 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. | DOI | MR | Zbl

[26] W.-M. Ni - I. Takagi - E. Yanagida, Stability analysis of point-condensation solutions to a reaction-diffusion system. Tokoku Math. J., submitted.

[27] W.-M. Ni - I. Takagi - E. Yanagida, Stability of least-energy patterns in a shadow system of an activator-inhibitor model. Japan J. Industrial Appl. Math., 18, 2001, 259-272. | DOI | MR | Zbl

[28] M.A. Pozio - A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system. Nonlinear Analysis, 14, 1990, 657-689. | DOI | MR | Zbl

[29] M.A. Pozio - A. Tesei, Invariant rectangles and strongly coupled semilinear paraboli systems. Forum Math., 2, 1990, 175-202. | fulltext EuDML | DOI | MR | Zbl

[30] D.H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J., 21, 1972, 979-1000. | MR | Zbl

[31] N. Shigesada - K. Kawasaki - E. Teramoto, Spatial segregation of interacting species. J. Theo. Biology, 79, 1979, 83-99. | DOI | MR

[32] S.-A. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Eqns., 185, 2002, 281-305. | DOI | MR | Zbl

[33] G. Sweers, A sign-changing global minimizer on a convex domain. In: C. BANDLE - J. BEMELMANS - M. CHIPOT - M. GRÜTER - J. ST. JEAN PAULIN (eds.), Progress in Partial Differential Equations: Elliptic and Parabolic Problems. Pitman Research Notes in Math., 266, Longman, Harlow 1992, 251-258. | MR | Zbl

[34] I. Takagi, Point-condensation for a reaction-diffusion system. J. Diff. Eqns., 61, 1986, 208-249. | DOI | MR | Zbl

[35] A. Trembley, Mémoires pour servir à l’histoire d’un genre de polypes d’eau douce, à bras en forme de cornes. Leyden 1744.

[36] A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London, B, 237, 1952, 37-72.

[37] P.E. Waltman, Competition models in population biology. CBMS-NSF Conf. Ser. Appl. Math., 45, SIAM, Philadelphia 1983. | DOI | MR | Zbl

[38] A. Yagi, Global solution to some quasilinear parabolic system in population dynamics. Nonlinear Analysis, 21, 1993, 531-556. | DOI | MR | Zbl