I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \geq 0,x \geq 1$ is
$$
u_{tx} = - (1 - u)^n
$$
with boundary conditions
$$
\begin{cases}
u_t(x, 0) = 1 - x \\
u(x, 0) = 0 \\
u(1, t) = 0
\end{cases}
$$
with $n \geq 1$ an integer.