Single-point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equation in domains with non-constant curvature

TL;DR

This paper demonstrates the existence of boundary gradient blow-up solutions at a single point for the diffusive Hamilton-Jacobi equation in domains with non-constant boundary curvature, extending previous results to more general geometries.

Contribution

It proves the existence of single-point GBU solutions in a broad class of domains with non-constant boundary curvature, using boundary-fitted coordinates and auxiliary functions.

Findings

Single-point GBU solutions exist in domains with non-constant boundary curvature.

The method involves boundary-fitted curvilinear coordinates and auxiliary functions.

The analysis requires complex calculations with boundary-fitted coordinates.

Abstract

We consider the diffusive Hamilton-Jacobi equation $u_t - \Delta u = |\nabla u|^p$ 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.