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

Patrick W. Dondl; Michael Scheutzow

doi:10.3934/nhm.2012.7.137

Article Contents

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

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

Patrick W. Dondl^1, and
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: February 28, 2011

Revised: August 31, 2011

Published: February 2012

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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.

Keywords:

Interfaces,
heterogeneous media,
random media,
semilinear parabolic equations with randomness,
asymptotic behavior of non-negative solutions.

Mathematics Subject Classification: Primary: 35K58; Secondary: 35R60, 35B40, 60H15.

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References

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