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Asymptotic Self-Similarity in Diffusion Equations with Nonconstant Radial Limits at Infinity

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Abstract

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space \(\mathbb {R}^n\), where \(n \ge 2\), assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as \(|x| \rightarrow \infty \). Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference w between the solution and the self-similar profile, but also an antiderivative W obtained by solving a linear elliptic problem which involves w as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero.

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Acknowledgements

This project started more than 15 years ago, but was left aside for a long time due to other priorities. The untimely demise of Geneviève Raugel in spring 2019 stimulated us to finish it, as a tribute to our highly estimated colleague and friend. The authors are indebted to Marius Paicu for his active participation at the early stage of this project, and to Emmanuel Russ for constant help on many technical questions. All three authors were supported by the Project ISDEEC ANR-16-CE40-0013 of the French Ministry of Higher Education, Research and Innovation.

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Author notes

  1. Geneviève Raugel: (\(\dagger \) May 10, 2019).

Authors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes et CNRS, 100 rue des Maths, 38610, Gières, France

    Thierry Gallay & Romain Joly

  2. Département de Mathématiques, CNRS et Université Paris-Saclay, 91405, Orsay, France

    Geneviève Raugel

Authors
  1. Thierry Gallay
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  2. Romain Joly
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  3. Geneviève Raugel
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Correspondence to Thierry Gallay.

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Gallay, T., Joly, R. & Raugel, G. Asymptotic Self-Similarity in Diffusion Equations with Nonconstant Radial Limits at Infinity. J Dyn Diff Equat 34, 2593–2638 (2022). https://doi.org/10.1007/s10884-020-09897-6

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  • DOI: https://doi.org/10.1007/s10884-020-09897-6

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