Complete classification of gradient blow-up and recovery of boundary condition for the viscous Hamilton-Jacobi equation

We study the Cauchy-Dirichlet pbm for superquadratic viscous Hamilton-Jacobi eq. We give a complete classification, namely rates and space-time profiles, in 1d case when viscosity sol. undergo gradient blow-up (GBU) or recovery of boundary condition (RBC) at any time when such phenomenon occurs. These results can be modified in radial domains in general dimensions. Previously, upper and lower estimates of GBU or RBC rates were available only in special case when basic comparison principle can be used. Even for type II BU in other PDEs, as far as we know, there has been no complete classification except [50], in which the argument relies on features peculiar to chemotaxis syst. Whereas there are many results on construction of special type II BU sol. of PDEs with investigation of (in-)stability of bubble, determination of (in-)stability of space-time profile for general sol. has not been done. In this paper, we determine whether space-time profile for each sol. is stable or unstable. A key in our proofs is to focus on algebraic structure with respect to vanishing intersections with singular steady state. In turn, GBU and RBC rates and profiles, as well as their (in-)stability, can be completely characterized by the number of vanishing intersections. We construct special sol. in bounded and unbounded intervals in both GBU and RBC cases, based on methods from [29], and then 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 sol. with specific behaviors in intermediate and outer regions, far from bubble and the RBC point, plays essential role. The application of such techniques to viscosity sol. is completely new