# American Institute of Mathematical Sciences

July  2009, 23(3): 617-638. doi: 10.3934/dcds.2009.23.617

## Front propagation in a noisy, nonsmooth, excitable medium

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, United States 2 Department of Mathematics, Michigan State University, East Lansing, MI, 48824, United States

Received  October 2007 Revised  August 2008 Published  November 2008

We consider the impact of noise on the stability and propagation of fronts in an excitable media with a piece-wise smooth, discontinuous ignition process. In a neighborhood of the ignition threshold the system interacts strongly with noise, the front can loose monotonicity, resulting in multiple crossings of the ignition threshold. We adapt the renormalization group methods developed for coherent structure interaction, a key step being to determine pairs of function spaces for which the the ignition function is Frechet differentiable, but for which the associated semi-group, $S(t)$, is integrable at $t=0$. We parameterize a neighborhood of the front solution through a dynamic front position and a co-dimension one remainder. The front evolution and the asymptotic decay of the remainder are on the same time scale, the RG approach shows that the remainder becomes asymptotically small, in terms of the noise strength and regularity, and the front propagation is driven by a competition between the ignition process and the noise.
Citation: Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617
 [1] G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131 [2] Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241 [3] Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477 [4] Hasan Alzubaidi, Tony Shardlow. Interaction of waves in a one dimensional stochastic PDE model of excitable media. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1735-1754. doi: 10.3934/dcdsb.2013.18.1735 [5] Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437 [6] Monica De Angelis, Pasquale Renno. Asymptotic effects of boundary perturbations in excitable systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2039-2045. doi: 10.3934/dcdsb.2014.19.2039 [7] Doron Levy, Tiago Requeijo. Modeling group dynamics of phototaxis: From particle systems to PDEs. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 103-128. doi: 10.3934/dcdsb.2008.9.103 [8] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [9] Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089 [10] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [11] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [12] Maria Aguareles, Marco A. Fontelos, Juan J. Velázquez. The structure of the quiescent core in rigidly rotating spirals in a class of excitable systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1605-1638. doi: 10.3934/dcdsb.2012.17.1605 [13] Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020010 [14] Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042 [15] Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175 [16] G. A. Braga, Frederico Furtado, Jussara M. Moreira, Leonardo T. Rolla. Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients: Analytical results. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 699-715. doi: 10.3934/dcdsb.2007.7.699 [17] Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787 [18] Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125 [19] Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168 [20] Xinfu Chen, Carey Caginalp, Jianghao Hao, Yajing Zhang. Effects of white noise in multistable dynamics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1805-1825. doi: 10.3934/dcdsb.2013.18.1805

2018 Impact Factor: 1.143