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

2011/02/28 by Dondl, Patrick W, Scheutzow, Michael
#35K58 #35R60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1102.5691

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.

Related

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• Pinning of interfaces in random media
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• Pinning of interfaces in a random medium with zero mean