What kind of approaches can be used to study the following quasilinear parabolic pde for a scalar function $u=u(x,t)$ ? $$ u_t= u \ u_{x x} $$

The physical problem where this pde comes from dictates that the Cauchy problem
of interest corresponds to an initial condition which is a third-order polynomial with no-constant term
$$
u(x,0) = x \ (x^2-s x + p), \ \ \ x \in \mathbb{R}^+, \ \ \ s p \neq 0
$$
Note that $u(x,0)$ is initially $\propto x$, and that $u(x,t)$ will remain so during the evolution as long as the pde makes sense.
Indeed, and this is the crux of the matter, numerical experiments and heuristic arguments strongly indicate that there exists a *blow-up time* $0< T=T(s,p) < \infty$, where the solution explodes to infinity, with
$$
u(x,t) \sim \frac{f(x)}{T-t}, \ \ \ t \to T^-
$$

- Can one prove this, and relate $T$ and the amplitude $f(x)$ to the parameters $s,p$ entering the initial condition ?
- It is possible (and how !) to continue the solution
*after*the blow-up time ? - A colleague of mine suggested that I could exchange the role of the dependent and independent variables, and/or to look at a paper by Clarkson, Fokas, \& Ablowitz, "Hodograph transformations of linearizable partial differential equations", where they study similar equations and relate them to the Harry-Dym equation and "solvable" (by inverse scattering transform or Painlevé reductions) variants. Needless to say, the functional form of the terms in my pde do not fall under the conditions of applicability of Clarkson-fokas-Ablowitz' main theorem, while the hodograph transformation: $$ (t,x,u(x,t)) \longrightarrow (t,y=u,x=v(y,t)) $$ so that (using $u_x = 1/v_y$, $u_{x x}=-v_y^{-3}v_{y y}$, $u_t= -v_t/v_y$) $$ v_t + y (v_y^{-1})_y =0 $$ does not seem to give a pde much easier to solve (or to recognize, such as the porous medium equation)

**Edit**: an alternative perhaps preferable form (closer to the porous medium equation!)
is for the function $w=\frac{1}{2}{\rm Log}(u^2)$, which obeys
$$
w_t = \left(w_x e^w\right)_x
$$
The Lie-point symmetries of the latter are recalled in this review (section 5), but none would match the initial condition I have to deal with.