PDE subscript notation

Question

What does this notation mean in a PDE: $(u_xu_t)_{xx}$? For example, I have checked that $(u_xu_t)_x=u_{xx}u_{xt}$ so the product rule does not apply. However, it applies in the former case. What's the general rule?

Answer 1

It means that, for a function $u=u(x,t)$, you are computing the following quantity \begin{align*} \frac{\partial^2}{\partial x \partial x} \left( \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial t} \right) = \frac{\partial}{\partial x} \left( \frac{\partial^2 u}{\partial x \partial x}\cdot \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} \frac{\partial^2 u}{\partial x \partial t} \right) \\ = \frac{\partial^3 u}{\partial x \partial x \partial x} \cdot \frac{\partial u}{\partial t} + 2\frac{\partial^2 u}{\partial x \partial x} \cdot \frac{\partial^2 u}{\partial x \partial t} + \frac{\partial u}{\partial x} \cdot \frac{\partial^3 u}{\partial x \partial x \partial t} \end{align*}