The Profile of boundary gradient blowup for the diffusive Hamilton–Jacobi equation

We consider the diffusive Hamilton-Jacobi equationu(t) - Delta u = vertical bar del u vertical bar(p),with Dirichlet boundary conditions in two space dimensions, which is a typical model-case in the theory of parabolic PDEs and also arises in the Kardar-Parisi-Zhang 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 < p <= 3, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the precise final asymptotic profile, under the formu(y)(x, y, T) similar to dp [y + C vertical bar x vertical bar(2(p-1)/(p-2))](-1/(p-1)), as(x, y) -> (0, 0).Interestingly, this result displays a new phenomenon of strong anisotropy of the profile, quite different from what is observed in other blowup problems for nonlinear parabolic equations, with the exponents 1/(p - 1) in the normal direction y and 2/(p - 2) in the tangential direction x. Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation, whereas the normal profile remains self-similar.