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

TL;DR

This paper rigorously derives the limiting Hamilton-Jacobi equation for subdiffusive processes described by an age-structured model, overcoming key mathematical challenges related to memory effects and non-integrable stationary measures.

Contribution

It establishes uniform Lipschitz estimates and proves convergence to the viscosity solution of the Hamilton-Jacobi equation for subdiffusive models, advancing understanding of large-scale asymptotics.

Findings

Established uniform Lipschitz estimates for the rescaled equations.

Proved convergence to the viscosity solution of the Hamilton-Jacobi equation.

Addressed mathematical challenges due to memory effects and non-integrable measures.

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 $(t,x) \to (t/\varepsilon, x/\varepsilon)$ 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 $\varepsilon \to 0$. 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.