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Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $ p$-Laplacian diffusion


Authors: Amal Attouchi and Philippe Souplet
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
Published electronically: March 1, 2016
MathSciNet review: 3572260
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Abstract: We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $ u_t=\Delta _p u+\vert\nabla u\vert^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
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
Email: souplet@math.univ-paris13.fr

DOI: https://doi.org/10.1090/tran/6684
Received by editor(s): April 19, 2014
Received by editor(s) in revised form: January 31, 2015
Published electronically: March 1, 2016
Article copyright: © Copyright 2016 American Mathematical Society