Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains
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- 作者：Li Yuxiang and Philippe Souplet
- 刊名：Communications in Mathematical Physics
- 出版年：2010
- 出版时间：January, 2010
- 年：2010
- 卷：293
- 期：2
- 页码：499-517
- 全文大小：297.7 KB
文摘
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 W ¨¬ \mathbbR2{\Omega\subset \mathbb{R}^2} and on the initial data, we show that the gradient blow-up singularity occurs only at a single point x0 ? ?W{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 \mathbbRn{\mathbb{R}^n}, we also obtain results on nondegeneracy and localization of blow-up points.