Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients

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December 2010, 5(4): 745-763. doi: 10.3934/nhm.2010.5.745

Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients

Jérôme Coville ^1, , Nicolas Dirr ^2, and Stephan Luckhaus ^3,

1.	Equipe BIOSP, INRA Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France
2.	Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
3.	Mathematisches Institut der Universität Leipzig, PF 100920, Leipzig, Germany

Received October 2009 Revised April 2010 Published November 2010

Abstract
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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.

Keywords: Random obstacles, Qualitative behavior of parabolic PDEs with random coefficients, Interface evolution in Random media..

Mathematics Subject Classification: 35R60, 35B09, 82C4.

Citation: 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

References:

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[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.
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[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.
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show all references

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.

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