, Volume 293, Issue 2, pp 499-517

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.

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.