I have attempted to get an answer on the math.stackexchange but have not got any answer for a while. Thus, I am posting the question here. I am analyzing the following problem in order to establish that it is well-posed and state the regularity of the solution.

$$u_t=u_{xx}+\frac{1}{T-t}(x-y)u_y,\;u(0,x,y)=u_0(x,y),\; x,y\in R, \;t \in [0,T]$$

I think I have found a reference to show existence of that equation. However, for the uniqueness and continuous dependance I have difficulty due to "bad" behavior of the coefficient $\frac{1}{T-t}(x-y)$ as $t$ approaches $T$. Moreover, application of the Hormander theorem gives the regularity only inside of the domain. In fact, I can have this problem well-posed on any finite interval $[0,t_1]$ for $t_1 < T$, but not on the whole interval $[0,T]$.

I can do change of variables $z:=y(T-t)$ and $u(t,x,y)=v(t,x,z)$. Then, after application of the chain rule, the problem becomes much simpler to analyze:

$$v_t=v_{xx}+xv_z,\;v(0,x,z)=u_0(0,x,zT),\; x,z\in R, \;t \in [0,T]$$

and one might notice that $v(T,x,0)=u(T,x,y)$. This is where I am having some difficulty to continue. First, the second formulation satisfies the conditions for existence and uniqueness, does that imply the original problem has a unique solution as well? Second, the energy estimates are used to show continuous dependence on the initial data, do they imply the original problem is also continuously dependent on the initial data? Third, application of the Hormander condition gives the regularity on the entire time domain $[0,T]$, does that imply the regularity of the original problem? Basically, I want to know if it is equivalent to consider the alternative formulation in order to establish existence, uniqueness, continuous dependance and regularity of the original problem. Would like to hear any suggestions or answers to that question.