Analyses for a mathematical model of the pattern formation on shells of molluscs
type: Item
instance of: scholarly article
title: Analyses for a mathematical model of the pattern formation on shells of molluscs
zbMATH Open document ID: 0846.35056
DOI: 10.1007/BF02662496
author: Ming Mei
author: Yingkun Xiao
published in: Applied Mathematics. Series B (English Edition)
publication date: 1996-09-29T00:00:00Z
review text: The authors study, as a model of the pattern formation on the shell of molluscs, the Cauchy problem (for \(x\in \mathbb{R}^3\)) \[ a_t- D_a \Delta a= - \mu a+ {a^2 s\over 1+ \kappa a^2+ \rho_0},\;s_t- D_s \Delta s= - \gamma s+ \sigma- {a^2 s\over 1+ \kappa a^2+ \rho_0}, \] \[ a(x, 0)= a_0(x),\quad s(0, x)= s_0(x). \] The functions \(a(x, t)\), \(s(x, t)\) represent, respectively, the activator density and the substrate density, \(D_a> 0\), \(D_s> 0\) are respectively the diffusion coefficients of \(a(x, t)\) and \(s(x, t)\) and \(\rho\), \(\sigma\), \(\mu\), \(\gamma\), \(\kappa\) are some positive constants. The existence and uniqueness of a global smooth solution of this problem and its stability and time decay rate are studied.
reviewed by: Ion Onciulescu
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 850815
zbMATH Keywords: pattern formation
zbMATH Keywords: global smooth solution
zbMATH Keywords: time decay rate
MaRDI profile type: Publication
cites work: Nonconstant equilibrium solutions for a model of pattern formation on shells of molluscs
cites work: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation
cites work: Q3984785
cites work: Q3028764
description: scientific article; zbMATH DE number 850815