I asked this question before but did not have any luck with an answer. It might be a student level question but I need to understand that with possibly some help. I am considering the hyperbolic equation of the form $$u_t+\frac{x}{T-t}u_x=0$$ and some initial data $u(x,0)=u_0(x)$. I would like to claim that this is a well posed problem in $\mathbb{L}^2$ on $[0,T]$. For that existence, uniqueness and stability needs to be established. So I have the following steps to achieve that.
Step 1: there is a change of variables I can employ as $z=x*(T-t)$. Then for $v(z(t,x),t)=u(x,t)$ I can find $$u_t=v_t+v_z(-x),\; u_x=v_z(T-t).$$Plug them into original equation I obtain an equivalent formulation to the original problem(up to a change of variables): $$v_t=0,\; v(z,0)=u_0(z/T,0).$$ The alternative formulation is clearly trivially has a unique solution for all $t\in [0,T]$, thus I claim the original problem does as well. Moreover, $u(x,T)=v(0,T)=u_0(0,0)$, so even though it initially looks like a not defined p.d.e. at that time it does have a solution.
Step 2: show stability via energy methods. For an arbitrary hyperbolic equation $w_t-a(y)w_y=0, \; y \in [0,1]$: \begin{align*} \frac{d||w||^2}{dt}&=(w_t,w)+(w,w_t)=(a(y)w_y,w)+(w,a(y)w_y)\\ &=(a(y)w_y,w)-(w_y,a(y)w)-(w,a_y(y)w)+a(y)ww|^1_0\\ &\leq max_{y\in [0,1]}|a_y(y)|||w||^2+a(y)ww|^1_0 \end{align*} Thus, the energy estimate boils down to the bound of $a_y$ provided boundary conditions are bounded. I have to estimate the stability of the original equation for $u(x,t)$ and it is not equivalent to the stability of $v(z,t)$. But from the energy estimate above $a_x=\frac{1}{T-t}$ which is not bounded as $t \to T$. Does that imply that is not well posed or energy methods did not work? What are other approaches to prove stability here as I can explicitly find the solution, so it perfectly stable on $[0,T]$? In this case everything is solvable, but I am considering the general case where the trajectories do have this kind of asymptotic behavior.