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Existence of Traveling Waves of Invasion for Ginzburg–Landau-type Problems in Infinite Cylinders

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Abstract

We study a class of systems of reaction–diffusion equations in infinite cylinders which arise within the context of Ginzburg–Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non-linearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach.

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Authors and Affiliations

  1. CSI Mathematics Department, The City University of New York, 2800 Victory Boulevard, Staten Island, New York, NY, 10314, USA

    M. Lucia

  2. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    C. B. Muratov

  3. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    M. Novaga

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  1. M. Lucia
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  2. C. B. Muratov
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  3. M. Novaga
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Correspondence to M. Lucia.

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Communicated by F. Otto

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Lucia, M., Muratov, C.B. & Novaga, M. Existence of Traveling Waves of Invasion for Ginzburg–Landau-type Problems in Infinite Cylinders. Arch Rational Mech Anal 188, 475–508 (2008). https://doi.org/10.1007/s00205-007-0097-x

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