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 that of diffusion. 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. 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.
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