Complete Classification of Gradient Blow-Up and Recovery of Boundary Condition for the Viscous Hamilton-Jacobi Equation

It is known that the Cauchy-Dirichlet problem for the superquadratic viscous Hamilton-Jacobi equation ut-Delta u=|del u|(p) with p > 2, which has important applications in stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after the appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. The solutions eventually become classical again for large time, but in-between they may undergo losses and recoveries of boundary conditions at multiple times (as well as GBU at multiple times), hence forming a quite complex dynamics. In this paper, with the help of braid group theory combined with inner/outer region analysis, we give a complete classification of all viscosity solutions in one-dimensional case. Namely, we fully describe the (countably many, type II) rates and space-time profiles of gradient blow-up (GBU) or recovery of boundary condition (RBC), we characterize them at any time when such a phenomenon occurs, and we moreover determine whether the space-time profiles are stable or unstable. These results can be modified in radial domains in general dimensions. Even for type II blow-up in other PDEs, as far as we know, there has been no complete classification except Mizoguchi (Commun Pure Appl Math 75:1870-1886, 2022), in which the argument relies on features peculiar to chemotaxis system. Whereas there are many results on construction of special type II blow-up solutions of PDEs with investigation of stability/instability of bubble, determination of stability/instability of space-time profile for general solutions has not been done. A key in our proofs is to focus on braid group algebraic structure with respect to vanishing intersections with the singular steady state, as a GBU or RBC time is approached. In turn, the GBU and RBC rates and profiles, as well as their stability/instability, can be given a complete geometric characterization by the number of vanishing intersections. We construct special solutions in bounded and unbounded intervals in both GBU and RBC cases, based on methods from Herrero and Velazquez (Preprint, 1994), and then we apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space-time profile. In the process, careful construction of special solutions with specific behaviors in intermediate and outer regions, which is far from bubble and the RBC point, plays an essential role. The application of such techniques to viscosity solutions is completely new.