Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation

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.

[1]  V. Calvez,et al.  A kinetic eikonal equation , 2012, 1202.2342.

[2]  L. Evans,et al.  Partial Differential Equations , 1941 .

[3]  Joel Goldstein,et al.  Measurement of the ratios of branching fractions B(Bs0 → Ds- π+)/B(B0 → D-π+) and B(B+ → anti-D0π+)/B(B0 → D-π+) , 2007 .

[4]  Guy Barles,et al.  Hamilton-Jacobi equations : approximations, numerical analysis and applications : Cetraro, Italy 2011 , 2013 .

[5]  J. Ross,et al.  Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  The Solution to Subdiffusion-reaction Equation for the System with One Mobile and One Static Reactant , 2013 .

[7]  F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control , 2013 .

[8]  A. Nepomnyashchy,et al.  Fronts in anomalous diffusion–reaction systems , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  A. Nepomnyashchy,et al.  An exactly solvable model of subdiffusion–reaction front propagation , 2013 .

[10]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[11]  B. Perthame.  Transport Equations in Biology , 2006 .

[12]  V. Méndez,et al.  Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities , 2010 .

[13]  S. Fedotov,et al.  Subdiffusive master equation with space-dependent anomalous exponent and structural instability. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  William Feller,et al.  On the Integral Equation of Renewal Theory , 1941 .

[15]  Werner Horsthemke,et al.  Kinetic equations for reaction-subdiffusion systems: derivation and stability analysis. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  N. Caillerie.  Large deviations of a velocity jump process with a Hamilton-Jacobi approach , 2016, 1602.07216.

[17]  I. Sokolov,et al.  Asymptotic front behavior in an A + B → 2A reaction under subdiffusion. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  J-F Mercure,et al.  Quantum oscillations in the anomalous phase in Sr3Ru2O7. , 2009, Physical review letters.

[19]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[20]  B. Perthame,et al.  Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait , 2018, 2004.07490.

[21]  E. Grenier,et al.  Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations , 2016 .

[22]  E. Montroll.  Random walks on lattices , 1969 .

[23]  P. Vigneaux.  Méthodes Level Set pour des problèmes d'interface en microfluidique , 2007 .

[24]  M. Freidlin,et al.  Geometric optics approach to reaction-diffusion equations , 1986 .

[25]  E. Cox,et al.  Physical nature of bacterial cytoplasm. , 2006, Physical review letters.

[26]  A. Nepomnyashchy.  Mathematical Modelling of Subdiffusion-reaction Systems , 2016 .

[27]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  C. O’Hern,et al.  The Bacterial Cytoplasm Has Glass-like Properties and Is Fluidized by Metabolic Activity , 2014, Cell.

[29]  M. Dahan,et al.  Single-molecule tracking in live cells reveals distinct target-search strategies of transcription factors in the nucleus , 2014, eLife.

[30]  Vicenç Méndez,et al.  Reaction–Transport Systems , 2010 .

[31]  S. M'el'eard,et al.  Singular Limits for Reaction-Diffusion Equations with Fractional Laplacian and Local or Nonlocal Nonlinearity , 2014, 1405.4746.

[32]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[33]  Enrico Gratton,et al.  Probing short-range protein Brownian motion in the cytoplasm of living cells , 2014, Nature Communications.

[34]  Andreas Hellander,et al.  Mesoscopic Modeling of Stochastic Reaction-Diffusion Kinetics in the Subdiffusive Regime , 2015, Multiscale Model. Simul..

[35]  S L Wearne,et al.  Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  T. Franosch,et al.  Anomalous transport in the crowded world of biological cells , 2013, Reports on progress in physics. Physical Society.

[37]  P. Souganidis,et al.  A PDE approach to geometric optics for certain semilinear parabolic equations , 1989 .

[38]  S. Mirrahimi,et al.  A Hamilton-Jacobi approach for a model of population structured by space and trait , 2013, 1307.8332.

[39]  Hugues Berry,et al.  Quantitative Convergence Towards a Self-Similar Profile in an Age-Structured Renewal Equation for Subdiffusion , 2015, 1503.08552.

[40]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[41]  Y. Garini,et al.  Transient anomalous diffusion of telomeres in the nucleus of mammalian cells. , 2009, Physical review letters.

[42]  I. Sokolov,et al.  Front propagation in A+B-->2A reaction under subdiffusion. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  S B Yuste,et al.  Subdiffusion-limited A+A reactions. , 2001, Physical review letters.

[44]  S. Fedotov,et al.  Nonlinear degradation-enhanced transport of morphogens performing subdiffusion. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.