2019/02/08 by Carlos Esteve, Esteve, Carlos · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1902.03080
openalex publication_date 2019/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the diffusive Hamilton-Jacobi equation ut - \Δ u = |\∇ν|p in a bounded planar domain with zero Dirichlet boundary condition. It is\nknown that, for p>2, the solutions to this problem can exhibit gradient\nblow-up (GBU) at the boundary. In this paper we study the possibility of the\nGBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph.\nSouplet, 2009], it was shown that single point GBU solutions can be constructed\nin very particular domains, i.e.~locally flat domains and disks. Here, we prove\nthe existence of single point GBU solutions in a large class of domains, for\nwhich the curvature of the boundary may be nonconstant near the GBU point.\n Our strategy is to use a boundary-fitted curvilinear coordinate system,\ncombined with suitable auxiliary functions and appropriate monotonicity\nproperties of the solution. The derivation and analysis of the parabolic\nequations satisfied by the auxiliary functions necessitate long and technical\ncalculations involving boundary-fitted coordinates.\n