I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line: $$ \begin{cases} u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x = u , & \quad t > 0, \, x = 0, \\[2mm] u|_{t=0} = u_0, & \quad t = 0, \, x > 0, \end{cases} \label{1}\tag{$\star$} $$
where $u_0$ is a nonnegative, bounded function with compact support on $\mathbb{R}_+^*$.
My ultimate goal is to show the $L^1$-convergence as $t\to\infty$ of $u(t,\cdot)$ towards the exponential profile $U : x \mapsto \lambda e^{-x}$, where $\lambda$ is determined by the mass of the initial data $u_0$ (note that the evolution given by \eqref{1} preserves the mass of $u_0$ over time).
So far, I have used a semigroup approach, treating the term $\frac{1}{1+t}(xu)_x$ as a vanishing source term, regarded as a perturbation of the simpler case without $\frac{1}{1+t}(xu)_x$.
This method works provided we have the following uniform upper bound on the first moment of $u$:
$$ \boxed{\exists C > 0 \text{ such that } \forall t >0, \quad J(t) := \int_{x=0}^{\infty} xu(t,x)dx < C} $$
I am confident that this statement holds, due to the compact support of the initial data, but I am currently unable to rigorously establish such an estimate.
I would greatly appreciate any insights or ideas to help derive this bound!
