Kelkel, Jan; Surulescu, Christina A weak solution approach to a reaction-diffusion system modeling pattern formation on seashells. (English) Zbl 1180.35279 Math. Methods Appl. Sci. 32, No. 17, 2267-2286 (2009). Summary: We investigate a reaction-diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Cited in 9 Documents MSC: 35K51 Initial-boundary value problems for second-order parabolic systems 35K57 Reaction-diffusion equations 35B36 Pattern formations in context of PDEs 35K58 Semilinear parabolic equations Keywords:global weak solution; iterative approach PDF BibTeX XML Cite \textit{J. Kelkel} and \textit{C. Surulescu}, Math. Methods Appl. Sci. 32, No. 17, 2267--2286 (2009; Zbl 1180.35279) Full Text: DOI References: [1] Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B 237 pp 37– (1952) · Zbl 1403.92034 [2] Gierer, A theory of biological pattern formation, Kybernetik 12 pp 30– (1972) [3] Segel, Dissipative structure: an explanation and an ecological example, Journal of Theoretical Biology 37 pp 545– (1972) [4] Meinhardt, A model for pattern formation on the shells of molluscs, Journal of Theoretical Biology 126 pp 63– (1987) · Zbl 0625.92002 [5] Meinhardt, The Algorithmic Beauty of Seashells (2003) [6] Wei, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, Journal of Nonlinear Science 11 pp 415– (2001) · Zbl 1141.35345 [7] Wei, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, Journal of Differential Equations 178 pp 478– (2002) · Zbl 1042.35005 [8] Granero-Porati, Temporal organization in a morphogenetic field, Journal of Mathematical Biology 20 pp 153– (1984) · Zbl 0549.92001 [9] Rothe, Lecture Notes in Mathematics, in: Global Solutions of Reaction-Diffusion Systems (1984) · Zbl 0546.35003 [10] Jiang, Global existence of solutions of an activator-inhibitor system, Discrete and Continuous Dynamical Systems 14 pp 737– (2006) · Zbl 1101.35046 [11] Murray, Mathematical Biology II (2003) [12] Evans, Partial Differential Equations (1998) · Zbl 0902.35002 [13] Smoller, Shock Waves and Reaction-Diffusion Equations (1994) · Zbl 0807.35002 [14] Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, Journal of Mathematical Biology 34 pp 148– (1995) · Zbl 0835.92006 [15] Surulescu, On the stationary interaction of a Navier-Stokes fluid with an elastic tube wall, Applicable Analysis 86 pp 149– (2007) · Zbl 1114.74016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.