Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients
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Mathematical Sciences, Durham University, Science Site, South Road, Durham DH1 3LE
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2.
Fakultät II, Institut für Mathematik, Sekr. MA 7–5, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin
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Received:
01 March 2011
Revised:
01 September 2011
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Primary: 35K58; Secondary: 35R60, 35B40, 60H15.
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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[J]. Networks and Heterogeneous Media, 2012, 7(1): 137-150. doi: 10.3934/nhm.2012.7.137
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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.
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