The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation

We consider the diffusive Hamilton-Jacobi equation \[u_t-\Delta u=|\nabla u|^p,\] with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For \(p>2\), solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range \(2