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arXiv · 1609.06933

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

Vincent Calvez·Pierre Gabriel·Álvaro Mateos González
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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.

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Vincent Calvez, Pierre Gabriel, Álvaro Mateos González. 2016-09-22. Limiting Hamilton-Jacobi Equation for the Large Scale Asymptotics of a Subdiffusion Jump-Renewal Equation. https://doi.org/10.3233/asy-191528

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