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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-2010-04987-6
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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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
Email: nalikako@math.uoa.gr

Nikolaos I. Katzourakis
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 11584, Athens, Greece
Email: nkatzourakis@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-2010-04987-6
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.