We consider the diffusive Hamilton-Jacobi equation ut - Delta u =vertical bar del
u vertical bar(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
boundaryfitted coordinates. (C) 2019 Elsevier Masson SAS. All rights reserved.