Slow motion for equal depth multiple-well gradient systems: The degenerate case

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January 2013, 33(1): 67-87. doi: 10.3934/dcds.2013.33.67

Slow motion for equal depth multiple-well gradient systems: The degenerate case

Fabrice Bethuel ^1, and Didier Smets ^1,

1.	UPMC-Paris6, UMR 7598 LJLL, Paris, F-75005, France, France

Received May 2011 Revised December 2011 Published September 2012

Abstract
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We extend the study [1] of gradient systems with equal depth multiple-well potentials to the case when some of the wells are degenerate, in the sense that the Hessian is non positive at those wells. The exponentially small speed, in terms of distances between fronts, typical of non degenerate potentials is replaced by an algebraic upper bound, whose degree depends on the degeneracy of the wells.

Keywords: Front propagation, gradient systems, multiple-well potential..

Mathematics Subject Classification: 35K57, 35B25, 35B3.

Citation: Fabrice Bethuel, Didier Smets. Slow motion for equal depth multiple-well gradient systems: The degenerate case. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 67-87. doi: 10.3934/dcds.2013.33.67

References:

[1]	F. Bethuel, G. Orlandi and D. Smets, Slow motion for gradient systems with equal depth multiple-well potentials, J. Differential Equations, 250 (2011), 53-94.
[2]	L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804.
[3]	J. Carr and R. L. Pego, Metastable patterns in solutions of u _t=ε² $u_{x x}$ - f（u） Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.
[4]	X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437. doi: 10.1016/j.jde.2004.05.017.
[5]	F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420. doi: 10.1016/j.jde.2007.03.007.

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References:

[1]	F. Bethuel, G. Orlandi and D. Smets, Slow motion for gradient systems with equal depth multiple-well potentials, J. Differential Equations, 250 (2011), 53-94.
[2]	L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804.
[3]	J. Carr and R. L. Pego, Metastable patterns in solutions of u _t=ε² $u_{x x}$ - f（u） Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.
[4]	X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437. doi: 10.1016/j.jde.2004.05.017.
[5]	F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420. doi: 10.1016/j.jde.2007.03.007.

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