I have the following PDE'S $$\frac{\partial}{\partial t}u(t,r)=\frac{\partial}{\partial r}\big((r-1/2)u(t,r)\big)$$ with $u(0, r)=u_0(r)$ compact supported in the interval $[0,1]$ and such that $\int_0^1u_0(r)dr=1$.
I know the solution explicitly, Through the characteristic method we know that that $$u(t,r)=u_0\big((r-1/2)e^t+1/2\big)e^t$$.
I am wondering about the $\lim_{t\to +\infty}\int_0^1u(t,r)G(r) dr$, with $G\in C([0,1], \mathbb R)$ a test function.
Since I know the explicit solution it is easy to understand that $$\lim_{t\to +\infty}\int_0^1u(t,r)G(r) dr=G(1/2)$$.
My question is, could I deduce something about the asymptotic behaviour without knowning the explicit solution but just looking at the expression of the PDE?
This is really closely related to the method of characteristics, but doesn't explicitly use the solution.
Consider $$J = J(a(t), b(t), t) = \int_{a(t)}^{b(t)} u(t,r)\; dr $$ where $a(t)$ and $b(t)$ are continuously differentiable functions. By the Fundamental Theorem of Calculus we have $$ \eqalign{\dfrac{dJ}{dt} &= b'(t) u(t,b(t)) - a'(t) u(t,a(t)) + \int_{a(t)}^{b(t)} \dfrac{\partial u}{\partial t}(t,r) \; dr \cr &= b'(t) u(t,b(t)) - a'(t) u(t,a(t)) + \int_{a(t)}^{b(t)} \dfrac{\partial}{\partial t} \left((r-1/2) u(t,r)\right)\; dr \cr &= (b'(t) + b(t) - 1/2) u(t, b(t)) - (a'(t) + a(t) - 1/2) u(t, a(t))\cr} $$ In particular, this is $0$ if $b(t)$ and $a(t)$ are solutions of the differential equation $x' + x - 1/2 = 0$. Those solutions all converge to $1/2$ as $t \to \infty$. Thus if $[a(0), b(0)]$ contains the support of $u_0$, for large $t$ the support of $u(t,\cdot)$ is contained in the small interval $[a(t), b(t)]$ near $1/2$, and $\int_{a(t)}^{b(t)} u(t,r)\; dr = 1$. Take $\epsilon > 0$. If the interval is small enough that $|G(r) - G(1/2)|< \epsilon$, we have
$$\left|\int_0^1 u(t,r) G(r)\; dr - G(1/2)\right| = \left|\int_{a(t)}^{b(t)} u(t,r) (G(r) - G(1/2))\; dr \right| < \epsilon \int_{a(0)}^{b(0)} |u(t,r)|\; dr $$
Thus we get $$ \int_0^1 u(t,r) G(r)\; dr \to G(1/2)\ \text{as}\ t \to \infty$$