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Heteroclinic travelling waves of gradient diffusion systems

Authors: Nicholas D. Alikakos and Nikolaos I. Katzourakis
Journal: Trans. Amer. Math. Soc. 363 (2011), 1365-1397
MSC (2010): Primary 35C07
Published electronically: October 22, 2010
MathSciNet review: 2737269
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Abstract: We establish the existence of travelling waves to the gradient system $ u_t = u_{zz} - \nabla W(u)$ connecting two minima of $ W$ when $ u : \mathbb{R} \times (0,\infty) \longrightarrow \mathbb{R}^N$; that is, we establish the existence of a pair $ (U,c) \in [C^2(\mathbb{R})]^N \times (0,\infty)$, satisfying

$\displaystyle \left\{\begin{array}{l} U_{xx} - \nabla W ( U ) = - c\; U_x,\\ U(\pm \infty) = a^{\pm}, \end{array}\right. $

where $ a^{\pm}$ are local minima of the potential $ W \in C_{\textrm{loc}}^2(\mathbb{R}^N)$ with $ W(a^-)< W(a^+)=0$ and $ N \geq 1$. Our method is variational and based on the minimization of the functional $ E_c (U) = \int_{\mathbb{R}}\Big\{ \frac{1}{2}\vert U_x\vert^2 + W( U ) \Big\}e^{cx} dx$ in the appropriate space setup. Following Alikakos and Fusco (2008), we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed.

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  • [A-Ba-C] N. Alikakos, P. Bates, X. Chen, Periodic travelling waves and oscillating patterns in multidimensional domains, Transactions of the A.M.S., Vol. 351, Nr. 7, (1999), 2777-2805. MR 1467460 (99j:35101)
  • [A-Be-C] N. Alikakos, S. Betelú, X. Chen, Explicit Stationary Solutions in Multiple Well Dynamics and Non-uniqueness of Interfacial Energy Densities, Euro. Jnl. of Applied Math. (2006), 17, 525-556. MR 2296027 (2007k:35119)
  • [A-F] N. Alikakos, G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. J. of Math., Vol. 57, No. 4, 1871 - 1906, (2008). MR 2440884 (2009h:37124)
  • [C et al.] X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya, J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 369-393. MR 2319939 (2009g:35063)
  • [DC] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976 (25th printing). MR 0394451 (52:15253)
  • [Ev] L. C. Evans, Partial Differential Equations, A.M.S., Graduate Texts in Mathematics, Vol. 19, 1998. MR 1625845 (99e:35001)
  • [F] P. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rat. Mech. Anal. 70 (1979), 31-46. MR 535630 (81f:35063)
  • [F-McL] P. Fife, J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65 (1977), 335-361. MR 0442480 (56:862)
  • [F-McL2] P. Fife, J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rat. Mech. Anal. 75 (1981), 281-314. MR 607901 (83b:35085)
  • [G-R] T. Gallay, E. Risler, A variational proof of global stability for bistable travelling waves, Diff. and Int. Equations 20 (2007), 901-926. MR 2339843 (2008m:35196)
  • [G-H-L] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer-Verlag, 1993, 2nd printing.
  • [G-T] D. Gilbarg, N. Trudinger, Elliptic Partial Differenial Equations of Second Order, Springer, 1998, revised 3rd edition.
  • [Hei] S. Heinze, Travelling Waves for Semilinear Parabolic Partial Differential Equations in Cylindrical Domains, Ph.D. thesis, Heidelberg University, 1988.
  • [H-P-S] S. Heinze, G. Papanicolaou, A. Stevens, Variational principles for propagation speeds in inhomogeneous media, SIAM J. of Appl. Math. (2001), Vol. 62, No. 1, 129-148. MR 1857539 (2002j:35169)
  • [He] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, 1981. MR 610244 (83j:35084)
  • [K-S] A. Kufner, A.-M. Sändig, Some Application of Weighted Sobolev Spaces, Leipzig, Teubner-Texte zur Mathematik, 1987. MR 926688 (89h:35096)
  • [LMN] M. Lucia, C. Muratov and M. Novaga, Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders, Arch. Rat. Mech. Anal., vol. 188, n 3, 475-508, 2008. MR 2393438 (2009i:35181)
  • [M] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Disc. Cont. Dyn. Syst. Ser B 4 (2004), 867-892. MR 2082914 (2005j:35124)
  • [R] E. Risler, Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure, Annales de l'Institut Henri Poincaré (C) Anal. Non Linéaire Analysis, Vol. 25, Issue 2, 381-424 (2008). MR 2400108 (2009b:35202)
  • [Stef] V. Stefanopoulos, Heteroclinic connections for multiple-well potentials: The anisotropic case, Proceedings of the Royal Society of Edinburgh, 138A, 13131330, 2008. MR 2488061 (2009m:37058)
  • [St] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems, Rocky Mountain J. of Math., 21, (1991), no. 2, 799-807. MR 1121542 (92e:49016)
  • [V] A. Volpert, V. Volpert, V. Volpert, Traveling wave solutions of parabolic systems, A.M.S., Translations of Mathematical Monographs Vol. 140, 1994. MR 1297766 (96c:35092)

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Additional Information

Nicholas D. Alikakos
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 11584, Athens, Greece – and – Institute of Applied and Computational Mathematics, Foundation for Research and Technology, GR 70013 Heraklion, Crete, Greece

Nikolaos I. Katzourakis
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 11584, Athens, Greece

Keywords: Gradient diffusion systems, parabolic PDEs, travelling waves, heteroclinic connections
Received by editor(s): December 24, 2007
Received by editor(s) in revised form: November 30, 2008
Published electronically: October 22, 2010
Additional Notes: The first author was partially supported by Kapodistrias Grant No. 70/4/5622 at the University of Athens.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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