Suppose that $w: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is a solution to the PDE $$\sum_{i=1}^n \frac{\partial^2 w}{\partial t_i^2} = c^2\sum_{i=1}^n \frac{\partial^2 w}{\partial x_i^2}.$$ Is there a name for this PDE? It isn't just the multi-dimensional wave equation as it has an equal number of time and space dimensions.
Furthermore are general solutions to the equation given by $F,G : \mathbb{R}^n \to \mathbb{R}$ so that $$w(t, x) = F(t -cv) +G(t+cv)$$ Finally what if there are more time dimensions than space dimensions and visa-versa.
