Limiting Hamilton-Jacobi Equation for the Large Scale Asymptotics of a Subdiffusion Jump-Renewal Equation

Abstract

Subdiffusive motion takes place at a much slower timescale than diffusive motion. As a preliminary step to studying reaction-subdiffusion pulled fronts, we consider here the hyperbolic limit $(t,x)\to (t/\epsilon ,x/\epsilon )$ 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 $\epsilon \to 0$. In this work we derive uniform Lipschitz estimates, and establish the convergence towards the viscosity solution of the limiting Hamilton-Jacobi equation. The two main obstacles overcome in this work are the non-existence of an integrable stationary measure, and the importance of memory terms in subdiffusion.

References in corpus (4)

• Anomalous transport in the crowded world of biological cells
• Nonlinear degradation enhanced transport of morphogens performing subdiffusion
• Dynamics of concentration in a population model structured by age and a phenotypical trait
• Mesoscopic modeling of stochastic reaction-diffusion kinetics in the subdiffusive regime