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 $(t,x) \to (t/\varepsilon, x/\varepsilon)$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2016
- DOI:
- 10.48550/arXiv.1609.06933
- arXiv:
- arXiv:1609.06933
- Bibcode:
- 2016arXiv160906933C
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- Asymptotic Analysis, 2019, 115 (1-2), pp.63-94