We consider the diffusive Hamilton-Jacobi equation $u_t-\mathrm{\Delta }u=|\mathrm{\nabla }u{|}^p$ in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for $p>2$, the solutions to this problem can exhibit gradient blow-up (GBU) at the boundary. In this paper we study the possibility of the GBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph. Souplet, 2009], it was shown that single point GBU solutions can be constructed in very particular domains, i.e. locally flat domains and disks. Here, we prove the existence of single point GBU solutions in a large class of domains, for which the curvature of the boundary may be nonconstant near the GBU point.

Our strategy is to use a boundary-fitted curvilinear coordinate system, combined with suitable auxiliary functions and appropriate monotonicity properties of the solution. The derivation and analysis of the parabolic equations satisfied by the auxiliary functions necessitate long and technical calculations involving boundary-fitted coordinates.

我们考虑扩散哈密顿-雅各比方程 ${ü}_{Ť}-\mathrm{\Delta }ü=|\mathrm{\nabla }ü{|}^p$在Dirichlet边界条件为零的有界平面域中。众所周知，$p>2$，此问题的解决方案可能会在边界处出现梯度爆炸（GBU）。在本文中，我们研究了将GBU集减少到单个点的可能性。在以前的工作中[Y.-X. Li，Souplet，2009年]显示，可以在非常特殊的域（即本地平面域和磁盘）中构建单点GBU解决方案。在这里，我们证明了在一大类域中单点GBU解的存在，对于这些域，边界的曲率在GBU点附近可能不是恒定的。

我们的策略是使用边界拟合曲线坐标系，并结合适当的辅助函数和适当的解决方案的单调性。对辅助函数满足的抛物线方程的推导和分析，需要进行长且技术上涉及边界拟合坐标的计算。