Ni: Diffusion and cross-diffusion in pattern formation

Ni, Wei-Ming:
Diffusion and cross-diffusion in pattern formation
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 15 (2004), fasc. n.3-4, p. 197-214, (English)
pdf (284 Kb), djvu (229 Kb). | MR2148879 | Zbl 1162.35370

Sunto

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.

Referenze Bibliografiche

[1] H. AMANN, Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems. Diff. Integral Eqns., 3, 1990, 13-75. | MR 1014726 | Zbl 0729.35062

[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 1242579 | Zbl 0810.35037

[3] H. AMANN, Supersolution, monotone iteration and stability. J. Diff. Eqns., 21, 1976, 365-377. | MR 407451 | Zbl 0319.35039

[4] R.S. CANTRELL - C. COSNER, Spatial ecology via reaction-diffusion equations. John Wiley and Sons, 2003. | fulltext (doi) | MR 2191264 | Zbl 1087.92058

[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. | fulltext (doi) | MR 480282 | Zbl 0338.35055

[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. | fulltext (doi) | MR 2018876 | Zbl 1047.35054

[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. | fulltext (doi) | MR 1030101 | Zbl 0667.34072

[9] J.K. HALE - J.M. VEGAS, A nonlinear parabolic equation with varying domain. Arch. Rat. Mech. Anal., 86, 1984, 99-123. | fulltext (doi) | MR 751304 | Zbl 0569.35048

[10] G.E. HUTCHINSON, An introduction to population ecology. Yale University Press, New Haven, CT 1978. | MR 492532 | Zbl 0414.92026

[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. | fulltext (doi) | MR 1163435 | Zbl 0766.35029

[12] D. LE, Cross diffusion systems on $n$ spatial dimensional domains. Indiana Univ. Math. J., 51, 2002, 625-643. | fulltext (doi) | MR 1911048 | Zbl 1046.35061

[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. | fulltext (doi) | MR 1415047 | Zbl 0867.35032

[15] Y. LOU - W.-M. NI, Diffusion vs cross diffusion: An elliptic approach. J. Diff. Eqns., 154, 1999, 157-190. | fulltext (doi) | MR 1685622 | Zbl 0934.35040

[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. | fulltext (doi) | MR 1616969 | Zbl 0960.35049

[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. | fulltext (doi) | MR 2026204 | Zbl 1174.35360

[18] H. MATANO, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS, 15, 1979, 401-454. | fulltext mini-dml | fulltext (doi) | MR 555661 | Zbl 0445.35063

[19] W.-M. NI, Some aspects of semilinear ellitpic equations. Lecture Notes, National Tsinghua Univ., Hsinchu-Taiwan-China 1987. | Zbl 0676.35026

[20] W.-M. NI, Diffusion, cross-diffusion, and their spike-layer steady states. Notices of Amer. Math. Soc., 45, 1998, 9-18. | MR 1490535 | Zbl 0917.35047

[21] W.-M. NI - P. POLACIK - E. YANAGIDA, Montonicity of stable solutions in shadow systems. Trans. Amer. Math. Soc., 353, 2001, 5057-5069. | fulltext (doi) | MR 1852094 | Zbl 0981.35018

[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. | fulltext (doi) | MR 849484 | Zbl 0635.35031

[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. | fulltext (doi) | MR 1115095 | Zbl 0754.35042

[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 | fulltext (doi) | MR 1219814 | Zbl 0796.35056

[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. | fulltext (doi) | MR 1337211 | Zbl 0843.35006

[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. | fulltext (doi) | MR 1842911 | Zbl 1200.35172

[28] M.A. POZIO - A. TESEI, Global existence of solutions for a strongly coupled quasilinear parabolic system. Nonlinear Analysis, 14, 1990, 657-689. | fulltext (doi) | MR 1049787 | Zbl 0716.35034

[29] M.A. POZIO - A. TESEI, Invariant rectangles and strongly coupled semilinear paraboli systems. Forum Math., 2, 1990, 175-202. | fulltext EuDML | fulltext (doi) | MR 1042618 | Zbl 0701.35091

[30] D.H. SATTINGER, Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J., 21, 1972, 979-1000. | MR 299921 | Zbl 0223.35038

[31] N. SHIGESADA - K. KAWASAKI - E. TERAMOTO, Spatial segregation of interacting species. J. Theo. Biology, 79, 1979, 83-99. | fulltext (doi) | MR 540951

[32] S.-A. SHIM, Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Eqns., 185, 2002, 281-305. | fulltext (doi) | MR 1935640 | Zbl 1032.35090

[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 1194233 | Zbl 0789.35066

[34] I. TAKAGI, Point-condensation for a reaction-diffusion system. J. Diff. Eqns., 61, 1986, 208-249. | fulltext (doi) | MR 823402 | Zbl 0627.35049

[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. | fulltext (doi) | MR 778562 | Zbl 0572.92019

[38] A. YAGI, Global solution to some quasilinear parabolic system in population dynamics. Nonlinear Analysis, 21, 1993, 531-556. | fulltext (doi) | MR 1245865 | Zbl 0810.35046

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche