We have the Cauchy problem of the equation
$u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$
with some given smooth ($C^1$) function $g$ as initial value.
I want to check if the problem is well defined for each time.
We know that a problem is well defined if the solution exists, is unique and depends continuously on the data of the problem.
I have computed that the solution of the problem is $u(x,t)=g(xe^{-t}) e^{x(1-e^{-t})}$.
So we have that the problem is well-defined if the function $u$ that we found is the unique solution of the problem and if $u$ depends continuously on the data of the problem, right?
How can we deduce that there is no other solution except from $u$ ?
Is it implied that $u$ depends continuously on the data of the problem, since it contains $g$ ?
To know if the problem is well posed, you have to check the relation between the characteristic curves and the curve along wich the initial conditions are given. For the solution to be unique, the curve for the initial conditions have to be nowhere characteristic, so is, nowhere tangent to any chacteristic curve.
In this case the projection of the characteristics on the $t-x$ plane are $x=c_1e^t$ and the curve for the initial conditions is the line $t=0$. These curves are nowhere tangent each other and the curve for the initial data ($t=0$) intersects each one of the family of characteristics. So, for each value of $c_1=xe^{-t}$ we have one value of $c_2=e^{-c_1}g(c_1)$ to determine a single characteristic curve ($u(0,x)=g(x)=c_2e^x$ and $c_1=x$, so $c_2=e^{-x}g(x)$ and the equation follows). Then, the solution is constructed finding for each $c_1$, $c_2$, that depends on $g$, a continuous function by hypothesis, to form curves and then varying $c_1$ to form a surface with these curves. The solution is the one you wrote: $u(x,t)=g(xe^{-t}) e^{x(1-e^{-t})}$