Affiliations: [a] Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. E-mail: vincent.calvez@mat.cnrs.fr | [b] Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 avenue des États-Unis, F-78035 Versailles cedex, France. E-mail: pierre.gabriel@uvsq.fr | [c] Institut Montpellliérain Alexander Grothendieck (IMAG), UMR CNRS 5149, Université de Montpellier, 34090 Montpellier, France | [d] Institut des Sciences de l’Évolution de Montpellier (ISEM), UMR CNRS 5554, Université de Montpellier, 34095 Montpellier, France | [e] MISTEA, UMR CNRS 0729, INRA and SupAgro Montpellier, 34060 Montpellier, France. E-mail: alvaro.mateos-gonzalez@umontpellier.fr
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)→(t/ε,x/ε) 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 ε→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.
