The stabilizing effect of growth on pattern formation

Mauricio Labadie (1)

1 CAMS - Centre d'Analyse et de Mathématique sociales

Based on abundant numerical and experimental evidence, it has been conjectured that growth should have some kind of stabilising effect on pattern formation. In this paper we answer affirmatively this question: under an isotropic regime, growth shifts the eigenvalues of the reaction-diffusion system towards the left on the complex plane. Since the real parts of the eigenvalues are smaller, we can interpret this fact as a gain of stability. We also prove that growth enhances the possibility of a solution to be global: a local solution (i.e. defined up to a finite time) has more chances to be global (i.e. to exist for all times) on a growing manifold than on a fixed manifold. Moreover, if growth is fast enough we show that the solutions are always global. We illustrate this "anti-blow-up" effect with two scalar examples, for which there is blow-up on fixed domains. We show that on growing domains the blow-up occurs later than in fixed domains, and that if growth is fast enough then here is no blow-up. We finish with a discussion of the results, showing that the classical linear stability analysis for bifurcations apply to this framework, and pointing out the possible applications of our results to regulatory dynamics in pattern formation, embryogenesis and tumor growth.

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. 2010. ⟨hal-00194920v4⟩