Analyses for a mathematical model of the pattern formation on shells of molluscs
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.
scientific article; zbMATH DE number 850815