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Subdiffusive motion takes place at a much slower timescale than diffusivemotion. 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 subdiffusivemotion of, e.g., some protein inside a biological cell. Solutions of therescaled equations are known to satisfy a Hamilton-Jacobi equation in theformal limit $\varepsilon \to 0$. In this work we derive uniform Lipschitzestimates, and establish the convergence towards the viscosity solution of thelimiting Hamilton-Jacobi equation. The two main obstacles overcome in this workare the non-existence of an integrable stationary measure, and the importanceof memory terms in subdiffusion.

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