Transport equation $p_t - (xp)_x = 0$ for density of substance So I'm trying to do the following:

i) Solve
  $$p_t - (xp)_x = 0 \quad\text{for}\quad (t,x) \in (0, \infty ) \times \mathbb{R}$$
$$p(0,x) = {p_0}(x) \quad\text{for}\quad x \in \mathbb{R}, {p_0} \in C_0^1 ( \mathbb{R})$$
  ii) Calculate $$\lim_{t \to \infty} \int_{\mathbb{R}} \phi(x) p(t,x) dx \quad\text{for}\quad \phi \in C( \mathbb{R})$$
  iii) Assuming that $p_0 \geq 0$ and $p$ is the density of a substance at the time $t$. Interpet i ) and ii) from a physical perspective! Is the mathematical result of ii) the same as one would expect physically?

Ok, so I think I've managed to solve i) with the method of characteristics and got $p = e^t p_0(xe^t)$.
For ii), I was thinking you could substitute $u=x e^t$ to get
\begin{aligned}
\lim_{t \to \infty} \int_{\mathbb{R}} \phi(x) p(t,x) dx &= \lim_{t \to \infty} \int_{\mathbb{R}} \phi(u/e^t) p_0(u) dx  \\
&= \int_{\mathbb{R}} \lim_{t \to \infty} \phi(u/e^t) p_0(u) dx \\
&= \int_{\mathbb{R}} \phi (0) p_0(u)
\end{aligned} 
but I am not sure if it is finished or even correct.
I also need help with iii), I know that $p$ is the density and that the integral over the density can give you the volume, but I am not sure how to physically interpret the formulas in i) and ii).
 A: The PDE $p_t−(xp)_x=0$ in (i) rewrites as $p_t−xp_x=p$. Its solution obtained via the method of characteristics is indeed $p(t,x)=e^t\, p_0(x e^t)$. Setting $u = x e^t $, the integral in (ii) rewrites as
$$
\Phi(t) = \int_{\Bbb R}\phi(x)\, p(t,x)\,\text d x = \int_{\Bbb R}\phi(ue^{-t})\, p_0(u)\,\text d u
$$
which limit at infinity is
$$
\lim_{t\to {+\infty}}\Phi(t) = \phi(0)\int_{\Bbb R}p_0(u)\,\text d u \, .
$$
The PDE in (i) is a conservation law. If $p$ is the density of a given substance, then the total amount of substance $P(t) = \int_{\Bbb R} p(t,x)\,\text d x$ evolves as
$$
P'(t) = \int_{\Bbb R} \partial_t p(t,x)\,\text d x = [x\, p(t,x)]_{x=-\infty}^{x=+\infty} = 0\, ,
$$
i.e. $P(t)= P(0)= \int_{\Bbb R} p_0$. The product $-xp$ is the flux of this substance, where $-x$ is analogous to a transport velocity (cf. conservation of mass). When $t$ goes to infinity, the substance flows towards $x=0$ where the speed is zero, and it stays concentrated there. This can be illustrated by a plot of the characteristic curves $x=x_0 e^{-t}$ in the $x$-$t$ plane:

The quantity $\Phi$ in (ii) expresses the total amount of $\phi$ related to the substance over the whole domain $\Bbb R$. According to the flow of the substance, the quantity $\Phi$ equals $\phi(0)$ multiplied by the total amount of substance $P (0)$ as $t$ goes to infinity.
