Non-existence of positive stationary solutions for a class of
semi-linear PDEs with random coefficients
-
1.
Equipe BIOSP, INRA Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9
-
2.
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY
-
3.
Mathematisches Institut der Universität Leipzig, PF 100920, Leipzig
-
Received:
01 October 2009
Revised:
01 April 2010
-
-
35R60, 35B09, 82C44.
-
-
We consider a so-called random
obstacle model for the motion of a hypersurface through a field of
random obstacles,
driven by a constant driving field.
The resulting semi-linear parabolic PDE with random coefficients does not
admit a global nonnegative stationary solution,
which implies that an interface that was flat originally cannot get
stationary.
The absence of global stationary solutions is shown by
proving lower bounds on the growth of stationary solutions on
large domains with Dirichlet boundary conditions.
Difficulties arise because the random
lower order part of the equation cannot be bounded uniformly.
Citation: Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class ofsemi-linear PDEs with random coefficients[J]. Networks and Heterogeneous Media, 2010, 5(4): 745-763. doi: 10.3934/nhm.2010.5.745
Related Papers:
| [1] |
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
|
| [2] |
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
|
| [3] |
Luca Lussardi, Stefano Marini, Marco Veneroni .
Stochastic homogenization of maximal monotone relations and applications. Networks and Heterogeneous Media, 2018, 13(1): 27-45.
doi: 10.3934/nhm.2018002
|
| [4] |
Martin Heida, Benedikt Jahnel, Anh Duc Vu .
Regularized homogenization on irregularly perforated domains. Networks and Heterogeneous Media, 2025, 20(1): 165-212.
doi: 10.3934/nhm.2025010
|
| [5] |
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
|
| [6] |
Luis Caffarelli, Antoine Mellet .
Random homogenization of fractional obstacle problems. Networks and Heterogeneous Media, 2008, 3(3): 523-554.
doi: 10.3934/nhm.2008.3.523
|
| [7] |
Thomas Geert de Jong, Georg Prokert, Alef Edou Sterk .
Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions. Networks and Heterogeneous Media, 2025, 20(1): 1-14.
doi: 10.3934/nhm.2025001
|
| [8] |
Peter Bella, Arianna Giunti .
Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13(1): 155-176.
doi: 10.3934/nhm.2018007
|
| [9] |
Daniel Peterseim .
Robustness of finite element simulations in densely packed random particle composites. Networks and Heterogeneous Media, 2012, 7(1): 113-126.
doi: 10.3934/nhm.2012.7.113
|
| [10] |
Avner Friedman .
PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7(4): 691-703.
doi: 10.3934/nhm.2012.7.691
|
-
Abstract
We consider a so-called random
obstacle model for the motion of a hypersurface through a field of
random obstacles,
driven by a constant driving field.
The resulting semi-linear parabolic PDE with random coefficients does not
admit a global nonnegative stationary solution,
which implies that an interface that was flat originally cannot get
stationary.
The absence of global stationary solutions is shown by
proving lower bounds on the growth of stationary solutions on
large domains with Dirichlet boundary conditions.
Difficulties arise because the random
lower order part of the equation cannot be bounded uniformly.
References
|
[1]
|
S. Brazovsii and T. Nattermann, Pinning and sliding of driven elastic systems: From domain walls to charge density waves, Advances in Physics, 53 (2004), 177-252. doi: 10.1080/00018730410001684197
|
|
[2]
|
L. A. Caffarelli, P. E. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361. doi: 10.1002/cpa.20069
|
|
[3]
|
N. Dirr, G. Karali and N. K. Yip, Pulsating wave for mean curvature flow in inhomogeneous medium, European Journal of Applied Mathematics, 19 (2008), 661-699. doi: 10.1017/S095679250800764X
|
|
[4]
|
N. Dirr and N. K. Yip, Pinning and de-pinning phenomena in front propagation in heterogeneous media, Interfaces and Free Boundaries, 8 (2006), 79-109. doi: 10.4171/IFB/136
|
|
[5]
|
G. R. Grimmett and D. R. Stirzaker, "Probability and Random Processes," Oxford University Press, Oxford, 1992.
|
|
[6]
|
P.-L. Lions and P. E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 667-677. doi: 10.1016/j.anihpc.2004.10.009
|
|
[7]
|
L. Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure Appl. Math., 6 (1953), 167-177. doi: 10.1002/cpa.3160060202
|
|
[8]
|
J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2
|
-
-
This article has been cited by:
| 1.
|
Julien Brasseur, Jérôme Coville,
Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle,
2023,
35,
1040-7294,
237,
10.1007/s10884-021-09988-y
|
|
| 2.
|
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
|
|
| 3.
|
James P. Lee-Thorp, Alexander G. Shtukenberg, Robert V. Kohn,
Crystal Growth Inhibition by Mobile Randomly Distributed Stoppers,
2020,
20,
1528-7483,
1940,
10.1021/acs.cgd.9b01609
|
|
| 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.
|
Patrick W. Dondl, Michael Scheutzow,
Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients,
2012,
7,
1556-181X,
137,
10.3934/nhm.2012.7.137
|
|
| 6.
|
Patrick Dondl, Martin Jesenko, Michael Scheutzow,
Infinite pinning,
2022,
54,
0024-6093,
760,
10.1112/blms.12599
|
|
| 7.
|
Luca Courte, Kaushik Bhattacharya, Patrick Dondl,
Bounds on precipitate hardening of line and surface defects in solids,
2020,
71,
0044-2275,
10.1007/s00033-020-01327-3
|
|
| 8.
|
T. Bodineau, A. Teixeira,
Interface Motion in Random Media,
2015,
334,
0010-3616,
843,
10.1007/s00220-014-2152-4
|
|
| 9.
|
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
|
|
-
-
DownLoad: