Comptes Rendus
Partial Differential Equations, Probabilities
The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation
Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 931-938.

We study the asymptotics of Allen–Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i) the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii) the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.117
Classification: 60H15, 35D40

Pierre-Louis Lions 1; Panagiotis E. Souganidis 2

1 Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, and CEREMADE, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
2 Department of Mathematics University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2020__358_8_931_0,
     author = {Pierre-Louis Lions and Panagiotis E. Souganidis},
     title = {The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {931--938},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {8},
     year = {2020},
     doi = {10.5802/crmath.117},
     language = {en},
}
TY  - JOUR
AU  - Pierre-Louis Lions
AU  - Panagiotis E. Souganidis
TI  - The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 931
EP  - 938
VL  - 358
IS  - 8
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.117
LA  - en
ID  - CRMATH_2020__358_8_931_0
ER  - 
%0 Journal Article
%A Pierre-Louis Lions
%A Panagiotis E. Souganidis
%T The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation
%J Comptes Rendus. Mathématique
%D 2020
%P 931-938
%V 358
%N 8
%I Académie des sciences, Paris
%R 10.5802/crmath.117
%G en
%F CRMATH_2020__358_8_931_0
Pierre-Louis Lions; Panagiotis E. Souganidis. The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 931-938. doi : 10.5802/crmath.117. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.117/

[1] Matthieu Alfaro; Dimitra Antonopoulou; Georgia Karali; Hiroshi Matano Generation of fine transition layers and their dynamics for the stochastic Allen–Cahn equation (2018) (https://arxiv.org/abs/1812.03804)

[2] Guy Barles; Halil M. Soner; Panagiotis E. Souganidis Front propagation and phase field theory, SIAM J. Control Optimization, Volume 31 (1993) no. 2, pp. 439-469 | DOI | MR | Zbl

[3] Guy Barles; Panagiotis E. Souganidis A new approach to front propagation problems: theory and applications, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 3, pp. 237-296 | DOI | MR | Zbl

[4] Xinfu Chen Generation and propagation of interfaces for reaction-diffusion equations, J. Differ. Equations, Volume 96 (1992) no. 1, pp. 116-141 | DOI | MR | Zbl

[5] Lawrence C. Evans; Halil M. Soner; Panagiotis E. Souganidis Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., Volume 45 (1992) no. 9, pp. 1097-1123 | DOI | MR | Zbl

[6] Tadahisa Funaki Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin., Engl. Ser., Volume 15 (1999) no. 3, pp. 407-438 | DOI | MR | Zbl

[7] Nobuyuki Ikeda; Shinzo Watanabe Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland, 1992 | Zbl

[8] Pierre-Louis Lions; Panagiotis E. Souganidis (in preparation) | DOI | Numdam

[9] Pierre-Louis Lions; Panagiotis E. Souganidis Fully nonlinear stochastic partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 9, pp. 1085-1092 | DOI | MR | Zbl

[10] Pierre-Louis Lions; Panagiotis E. Souganidis Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications, C. R. Math. Acad. Sci. Paris, Volume 327 (1998) no. 8, pp. 735-741 | DOI | Zbl

[11] Pierre-Louis Lions; Panagiotis E. Souganidis Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 10, pp. 783-790 | DOI | MR | Zbl

[12] Takao Ohta; David Jasnow; Kyozi Kawasaki Universal scaling in the motion of random interfaces, Phys. Rev. Lett., Volume 49 (1982) no. 17, pp. 1223-1226 | DOI

[13] Panagiotis E. Souganidis Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular Random Dynamics (Lecture Notes in Mathematics), Volume 2253, Springer, 2019 | DOI | MR

[14] Nung K. Yip Stochastic motion of mean curvature, Arch. Ration. Mech. Anal., Volume 144 (1998) no. 4, pp. 313-355 | MR | Zbl

Cited by Sources:

Comments - Policy