Limiting HamiltonJacobi Equation for the Large Scale Asymptotics of a Subdiffusion JumpRenewal Equation
Abstract
Subdiffusive motion takes place at a much slower timescale than diffusive motion. As a preliminary step to studying reactionsubdiffusion pulled fronts, we consider here the hyperbolic limit $(t,x) \to (t/\varepsilon, x/\varepsilon)$ of an agestructured equation describing the subdiffusive motion of, e.g., some protein inside a biological cell. Solutions of the rescaled equations are known to satisfy a HamiltonJacobi 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 HamiltonJacobi equation. The two main obstacles overcome in this work are the nonexistence of an integrable stationary measure, and the importance of memory terms in subdiffusion.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 DOI:
 10.48550/arXiv.1609.06933
 arXiv:
 arXiv:1609.06933
 Bibcode:
 2016arXiv160906933C
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 Asymptotic Analysis, 2019, 115 (12), pp.6394