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Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

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Abstract

Consider the diffusive Hamilton-Jacobi equation u t = Δu + |∇u|p, p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂Ω. In this paper, under suitable assumptions on \({\Omega\subset \mathbb{R}^2}\) and on the initial data, we show that the gradient blow-up singularity occurs only at a single point \({x_0\in\partial\Omega}\). This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of \({\mathbb{R}^n}\), we also obtain results on nondegeneracy and localization of blow-up points.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, P. R. China

    Li Yuxiang

  2. CNRS UMR 7539, Laboratoire Analyse Géométrie et Applications, Université Paris 13, 99, Avenue J.B. Clément, 93430, Villetaneuse, France

    Li Yuxiang & Philippe Souplet

Authors
  1. Li Yuxiang
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  2. Philippe Souplet
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Correspondence to Philippe Souplet.

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Communicated by P. Constantin

Supported in part by National Natural Science Foundation of China 10601012 and Southeast University Award Program for Outstanding Young Teachers 2005.

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Yuxiang, L., Souplet, P. Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains. Commun. Math. Phys. 293, 499–517 (2010). https://doi.org/10.1007/s00220-009-0936-8

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  • DOI: https://doi.org/10.1007/s00220-009-0936-8

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