The stabilizing effect of growth on pattern formation

Mauricio Labadie (1)

1 CAMS - Centre d'Analyse et de Mathématique sociales

We want to understand the effect of growth on the stability of Turing patterns on a curved domain (i.e. a manifold), and in that spirit we prove three results. First, under the hypotheses of Fick's law of diffusion and conservation of mass we deduce a reaction-diffusion system on an arbitrary compact Riemannian manifold, generalizing the results of Plaza et al, "The effect of growth and curvature on pattern formation", J. Dyn. Differential Equations, 16 (4), 1093-1121. Second, we prove global existence, uniqueness and regularity for reaction-diffusion systems on a manifold with isotropic growth. Third, we show that under the isotropic regime the growth has both a "globalizing" and a "stabilizing" effect on pattern formation: a "globalizing effect" because on a growing manifold a locally defined solution (i.e. defined up to a finite time T) has more chances to be globally defined (i.e. for all times) than on a fixed manifold, and a "stabilizing effect" because the growth shifts the eigenvalues of the reaction-difusion system towards the left in the complex plane.

Mots clés: Stability; Pattern formation; Reaction-diffusion; Laplace-Beltrami operator; Global existence of solutions; Stability.

Domaines: Equations aux dérivées partielles [math.AP]; Formation de Structures et Solitons [nlin.PS]

Mauricio Labadie. The stabilizing effect of growth on pattern formation. 2008. ⟨hal-00194920v2⟩