I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am considering the hyperbolic equation of the form $$w_t+\frac{x}{T-t}w_x=0, x\in[0,1]$$ and some continuous initial data $w(x,0)=w_0(x)$. I would like to claim that this is a well posed problem on $[0,T]$. One can analytically solve this equation and obtain the unique solution but I am stuck with "continuous dependence" on the initial data. So I did the following. Choose $L^2$ space and use energy methods for an equation of the form $w_t-a(x)w_x=0, \; y \in [0,1]$: $$ \frac{d||w||^2}{dt}=(w_t,w)+(w,w_t)=(a(y)w_x,w)+(w,a(x)w_x) $$ $$ =(a(x)w_x,w)-(w_x,a(x)w)-(w,a_x(x)w)+a(x)ww|^1_0 $$ $$ \leq max_{x\in [0,1]}|a_x(x)|||w||^2+a(x)ww|^1_0 $$ Thus, the energy estimate boils down to the bound of $a_x$ provided boundary conditions are bounded. In my case $a(x)=-\frac{x}{T-t}$ and $a_x=\frac{1}{T-t}$. Thus, I can't show that inequality for any $t$ only for $t\in [0,T-\epsilon]$. However, I do know the solution at $t=T$ being $u(x,T)=u_0(0,0)$ and it is constant so the $L^2$ norm is perfectly bounded. Thus, please suggest me some ways to make a complete statement about this problem, is that well posed in $L^2$?

If I take $L^{\infty}$ instead to show such a dependence on initial data it directly follows from method of characteristics and clearly it is well posed in $L^{\infty}([0,T])$ but I am not sure I can have some "limiting statement" in $L^2$ as so far I have only on $[0,T-\epsilon]$