Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation.

Vincent Calvez (1, 2) , Pierre Gabriel (3) , Álvaro Mateos González (4, 1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 LMV - Laboratoire de Mathématiques de Versailles
4 BEAGLE - Artificial Evolution and Computational Biology

Subdiffusive motion takes place at a much slower timescale than that of diffusion. As a preliminary step to studying reaction-subdiffusion pulled fronts, we consider here the hyperbolic limit $(t,x) \to (t/\e, x/\e)$ of an age-structured equation describing the subdiffusive motion of, e.g., some protein inside a biological cell. Solutions of the rescaled equations are known to satisfy a Hamilton-Jacobi equation in the formal limit $\e \to 0$. In this work we derive uniform Lipschitz estimates, and establish the convergence towards the viscosity solution of the limiting Hamilton-Jacobi equation. Respectively, the two main obstacles overcome in the process are the non-existence of an integrable stationary measure, and the importance of memory terms in subdiffusion.

Mots clés: age-structured PDE - renewal equation - anomalous diffusion - large deviations - WKB approximation - Hamilton-Jacobi equation

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, Pierre Gabriel, Álvaro Mateos González. Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation.. 2016. ⟨hal-01372949v1⟩