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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion
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by Amal Attouchi and Philippe Souplet PDF
Trans. Amer. Math. Soc. 369 (2017), 935-974 Request permission

Abstract:

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta _p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion ($p=2$). The analysis in the case $p>2$ is considerably more delicate.
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Additional Information
  • Amal Attouchi
  • Affiliation: Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Sorbonne Paris Cité, CNRS (UMR 7539), 93430 Villetaneuse, France
  • MR Author ID: 986258
  • Email: attouchi@math.cnrs.fr
  • Philippe Souplet
  • Affiliation: Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Sorbonne Paris Cité, CNRS (UMR 7539), 93430 Villetaneuse, France
  • MR Author ID: 314071
  • Email: souplet@math.univ-paris13.fr
  • Received by editor(s): April 19, 2014
  • Received by editor(s) in revised form: January 31, 2015
  • Published electronically: March 1, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 935-974
  • MSC (2010): Primary 35B40, 35B45, 35K20, 35K92; Secondary 82C24, 35F21
  • DOI: https://doi.org/10.1090/tran/6684
  • MathSciNet review: 3572260