Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

Patrick W. Dondl; Michael Scheutzow; Patrick W. Dondl; Michael Scheutzow

doi:10.3934/nhm.2012.7.137

Networks and Heterogeneous Media

2012, Volume 7, Issue 1: 137-150. doi: 10.3934/nhm.2012.7.137

Previous Article Next Article

Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

Patrick W. Dondl ^{1
,},
Michael Scheutzow ^{2
,}

1.
Mathematical Sciences, Durham University, Science Site, South Road, Durham DH1 3LE
2.
Fakultät II, Institut für Mathematik, Sekr. MA 7–5, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin

Received: 01 March 2011 Revised: 01 September 2011
Primary: 35K58; Secondary: 35R60, 35B40, 60H15.

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.
- Interfaces,
- heterogeneous media,
- random media,
- semilinear parabolic equations with randomness,
- asymptotic behavior of non-negative solutions
Citation: Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients[J]. Networks and Heterogeneous Media, 2012, 7(1): 137-150. doi: 10.3934/nhm.2012.7.137

Related Papers:

Abstract

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.

References

[1]	S. Brazovskii and T. Nattermann, Pinning and sliding of driven elastic systems: From domain walls to charge density waves, Adv. Phys., 53 (2004), 177-252. Availabe from: arXiv:cond-mat/0312375.
[2]	J. Coville, N. Dirr and S. Luckhaus, Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients, Networks and Heterogeneous Media, 5 (2010), 745-763.
[3]	N. Dirr, P. W. Dondl, G. R. Grimmett, A. E. Holroyd and M. Scheutzow, Lipschitz percolation, Electron. Commun. Probab., 15 (2010), 14-21.
[4]	N. Dirr, P. W. Dondl and M. Scheutzow, Pinning of interfaces in random media, Interfaces and Free Boundaries, 13 (2011), 411-421. Available from: arXiv:0911.4254.
[5]	M. Kardar, Nonequilibrium dynamics of interfaces and lines, Phys. Rep., 301 (1998), 85-112. Available from: arXiv:cond-mat/9704172. doi: 10.1016/S0370-1573(98)00007-6
[6]	L. Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure Appl. Math., 6 (1953), 167-177.
[7]	D. Siegmund, On moments of the maximum of normed partial sums, Ann. Math. Statist., 40 (1969), 527-531. doi: 10.1214/aoms/1177697720

Reader Comments

Your name:*

Email:*
© 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)