March  2012, 7(1): 137-150. doi: 10.3934/nhm.2012.7.137

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

1. 

Mathematical Sciences, Durham University, Science Site, South Road, Durham DH1 3LE, United Kingdom

2. 

Fakultät II, Institut für Mathematik, Sekr. MA 7–5, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany

Received  March 2011 Revised  September 2011 Published  February 2012

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.
Citation: Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks and Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
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.

show all references

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.

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