PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\quad \forall (t,x)\in (0,1)^2$$
with the terminal condition $v(1,\cdot)=0$ and boundary conditions $v(\cdot,0)=v(\cdot,1)=-1$.
Has the wellposedness of this PDE been studied? Any answer, comments and related references are highly appreciated.
PS : Thanks to Zachary's observation, with the change of variable $-v:=\exp(-u)$, we obtain a new PDE
$$u_t(t,x) = -e^{u(t,x)}u_{xx}(t,x),\quad \forall (t,x)\in (0,1)^2,$$
with the terminal condition $u(1,\cdot)=\infty$ and boundary conditions $u(\cdot,0)=u(\cdot,1)=0$.
PS : Adopting terceira's suggestion by writing $v(t,x)=f(t)g(x)$, one has
$$f'(t)g(x)^3=g''(x)g(x)-g'(x)^2$$
and further
$$f'(t)\quad \equiv\quad \mbox{Constant}\equiv C \quad \equiv \quad \frac{g''(x)g(x)-g'(x)^2}{g(x)^3}.$$
Combining with the terminal condition one obtains $f(t)=C(t-1)$. How to deal with the ODE for $g$?
