I have difficulties understanding how to solve a PDE in $\mathbb{R}^{4}$ using the method of characteristics. I have a limited background in solving PDEs. I have seen only 2-dim examples and none for higher dimensions, and really don't understand how to generalize the method to the 4-dim case.

Let $f$ be a $C^{1}$ function on $\mathbb{R}^{4}$. The goal is to solve in general

$Xf=uf$, with $X=\sum_{i=1}^{4}X_{i}\frac{\partial}{\partial x_{i}}$ a smooth vector field and $u$ a continuous function on $\mathbb{R}^{4}$ (not explicitly given).

No initial data is given so I'm free to prescribe it myself.

*Attempt at the solution*:

I have four variables, so we consider the curve $(\gamma_{1}(t),\gamma_{2}(t),\gamma_{3}(t),\gamma_{4}(t),\gamma_{5}(t))$ in $\mathbb{R}^{5}$ such that we have five ODEs:

$\frac{d\gamma_{1}}{dt}=X_{1}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$

$\frac{d\gamma_{2}}{dt}=X_{2}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$

$\frac{d\gamma_{3}}{dt}=X_{3}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$

$\frac{d\gamma_{4}}{dt}=X_{4}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$

$\frac{d\gamma_{5}}{dt}=u$

(is this correct so far?)

Now, from this point, I don't understand *how and where to prescribe the initial values*. That is, according to the general theory (for an equation for $\mathbb{R}^{n}$) one has to pick any transversal hypersurface on which one has the initial values, and it has dimension $n-1$. So is it right that it's 3-dim in our case? But then one variable "falls out" since I have four variables. I really don't understand how, and exactly for what, to give initial values.

Could anyone clarify this please? And how do I proceed from there? I'd be very grateful not to receive references to look in books, since I've done so multiple times and the theory there doesn't help... Thank you.

This is not homework.