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Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

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Abstract

We study the long-time asymptotics of the doubly nonlinear diffusion equation \({\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}\) in \({\mathbb{R}^n}\), in the range \({\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}}\) and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \({\frac{2n}{n+1} < p < \frac{2n+1}{n+1}}\) where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, PO Box 3060, STN CSC, Victoria, BC, V8W 3R4, Canada

    Martial Agueh

  2. GREMAQ, Université de Toulouse, Manufacture des tabacs, 21 allée de Brienne, 31000, Toulouse, France

    Adrien Blanchet

  3. ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    José A. Carrillo

Authors
  1. Martial Agueh
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  2. Adrien Blanchet
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  3. José A. Carrillo
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Correspondence to Adrien Blanchet.

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Agueh, M., Blanchet, A. & Carrillo, J.A. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime. J. Evol. Equ. 10, 59–84 (2010). https://doi.org/10.1007/s00028-009-0040-8

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