Singularity formation for Burgers' equation with transverse viscosity

We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, (x,y)\in \mathbb R^2, u:[0,T)\times R^2\rightarrow R.$$ construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given backward self-similar along $x$ variable, whose scaling parameters evolve according to parabolic equations $y$ one them being quadratic semi-linear heat equation. develop new framework adapted this mixed hyperbolic/parabolic blow-up problem, revisit construction flat profiles for equation, self-similarity shocks