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
-
2.
Fakultät II, Institut für Mathematik, Sekr. MA 7–5, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin
-
Received:
01 March 2011
Revised:
01 September 2011
-
-
Primary: 35K58; Secondary: 35R60, 35B40, 60H15.
-
-
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
Related Papers:
| [1] |
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
|
| [2] |
Jérôme Coville, Nicolas Dirr, Stephan Luckhaus .
Non-existence of positive stationary solutions for a class of
semi-linear PDEs with random coefficients. Networks and Heterogeneous Media, 2010, 5(4): 745-763.
doi: 10.3934/nhm.2010.5.745
|
| [3] |
Clément Cancès .
On the effects of discontinuous capillarities for immiscible two-phase flows in
porous media made of several rock-types. Networks and Heterogeneous Media, 2010, 5(3): 635-647.
doi: 10.3934/nhm.2010.5.635
|
| [4] |
Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko .
Sharp interface limit in a phase field model of cell motility. Networks and Heterogeneous Media, 2017, 12(4): 551-590.
doi: 10.3934/nhm.2017023
|
| [5] |
Francesca R. Guarguaglini .
Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks and Heterogeneous Media, 2018, 13(1): 47-67.
doi: 10.3934/nhm.2018003
|
| [6] |
Mingming Fan, Jianwen Sun .
Positive solutions for the periodic-parabolic problem with large diffusion. Networks and Heterogeneous Media, 2024, 19(3): 1116-1132.
doi: 10.3934/nhm.2024049
|
| [7] |
Serge Nicaise, Cristina Pignotti .
Asymptotic analysis of a simple model of fluid-structure interaction. Networks and Heterogeneous Media, 2008, 3(4): 787-813.
doi: 10.3934/nhm.2008.3.787
|
| [8] |
Ken-Ichi Nakamura, Toshiko Ogiwara .
Periodically growing solutions in a class of strongly monotone semiflows. Networks and Heterogeneous Media, 2012, 7(4): 881-891.
doi: 10.3934/nhm.2012.7.881
|
| [9] |
Vincent Renault, Michèle Thieullen, Emmanuel Trélat .
Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12(3): 417-459.
doi: 10.3934/nhm.2017019
|
| [10] |
Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski .
An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12(1): 147-171.
doi: 10.3934/nhm.2017006
|
-
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.
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
|
-
-
This article has been cited by:
| 1.
|
T. Bodineau, A. Teixeira,
Interface Motion in Random Media,
2015,
334,
0010-3616,
843,
10.1007/s00220-014-2152-4
|
|
| 2.
|
Patrick W. Dondl, Michael Scheutzow,
Ballistic and sub-ballistic motion of interfaces in a field of random obstacles,
2017,
27,
1050-5164,
10.1214/17-AAP1279
|
|
| 3.
|
Patrick Dondl, Martin Jesenko, Michael Scheutzow,
Infinite pinning,
2022,
54,
0024-6093,
760,
10.1112/blms.12599
|
|
| 4.
|
Luca Courte, Patrick Dondl, Ulisse Stefanelli,
Pinning of interfaces by localized dry friction,
2020,
269,
00220396,
7356,
10.1016/j.jde.2020.06.005
|
|
| 5.
|
Luca Courte, Patrick Dondl, Michael Ortiz,
A Proof of Taylor Scaling for Curvature-Driven Dislocation Motion Through Random Arrays of Obstacles,
2022,
244,
0003-9527,
317,
10.1007/s00205-022-01765-5
|
|
| 6.
|
Fan Chen, Ming Cui, Chenguang Zhou,
A type of efficient multigrid method for semilinear parabolic interface problems,
2025,
143,
10075704,
108632,
10.1016/j.cnsns.2025.108632
|
|
-
-
DownLoad: