# 2nd order PDE with highest order term a mixed derivative

I stumbled upon an odd little pde (in the height $H$) here, and I don't really know where to begin the analysis.

$$0 = \frac{\partial^2 H }{\partial t \partial x } +\frac{\partial f}{\partial x}H + f \frac{\partial H}{\partial x} + H$$

where the prescribed function $f=f(x,t)$ is, unfortunately, time-dependent but otherwise can be taken to be sufficiently "nice". I also know $H(x,0) \geq 0$ and decreasing in $x$, and my intuition for the physical system tells me $H(x,t)$ should be non-negative and decreasing in $x$ for all time.

Is there hope for my little pde?

Make the change of variable $y = x-t$ and $z=x+t$, which is equivalent to $x = (y+z)/z$ and $t = (z-y)/2$. Then $$\frac{\partial}{\partial z} = \frac{1}{2} \frac{\partial}{\partial x} + \frac{1}{2} \frac{\partial}{\partial t}$$ and $$\frac{\partial}{\partial y} = \frac{1}{2} \frac{\partial}{\partial x} - \frac{1}{2} \frac{\partial}{\partial t}.$$ Then $$\frac{\partial^2}{\partial z^2} - \frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial x \partial t}.$$
In the new variables your problem becomes $$0 = \partial_z^2 H - \partial_y^2 H + H(\partial_z f +\partial_y f) + f(\partial_z H +\partial_y H ) +H,$$ and this is a wave equation with lower-order terms.