A stochastic control problem has led me to the following PDE:

State space: $t \in [0,T]$ and $x \in [-1,1]$.

$$4 \frac{\partial f}{\partial t} \frac{\partial^2 f}{\partial x^2} = 1 \quad \forall (t,x) \in [0,T) \times (-1,1)$$ and the boundary conditions are: $$ \begin{split} f(T,x) &= 0\quad \forall x\\ &\text{and} \\ f(t,\pm 1) &= T-t \end{split}$$ Is there an argument that can let me say that a solution exists to this PDE? I am not after a closed form expression obviously.

From the properties of the problem I can argue that $f(t,x) = f(t,-x)$ and $\arg \max_x f(t,x) = 0$. Don't know if it is of any use though.

Thanks.