# American Institute of Mathematical Sciences

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 & Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
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##### 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.   Google Scholar [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.   Google Scholar [3] N. Dirr, P. W. Dondl, G. R. Grimmett, A. E. Holroyd and M. Scheutzow, Lipschitz percolation,, Electron. Commun. Probab., 15 (2010), 14.   Google Scholar [4] N. Dirr, P. W. Dondl and M. Scheutzow, Pinning of interfaces in random media,, Interfaces and Free Boundaries, 13 (2011), 411.   Google Scholar [5] M. Kardar, Nonequilibrium dynamics of interfaces and lines,, Phys. Rep., 301 (1998), 85.  doi: 10.1016/S0370-1573(98)00007-6.  Google Scholar [6] L. Nirenberg, A strong maximum principle for parabolic equations,, Comm. Pure Appl. Math., 6 (1953), 167.   Google Scholar [7] D. Siegmund, On moments of the maximum of normed partial sums,, Ann. Math. Statist., 40 (1969), 527.  doi: 10.1214/aoms/1177697720.  Google Scholar
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