$e^{-d\tau}\theta(\tau,\eta) = u(t,x), \label{def:theta}$$ with coordinates $e^\tau\eta = x$ and $\beta e^{\beta^{-1}\tau} - \beta = t$, where $\beta$ is given by . In what follows we denote $\alpha := d\beta$. In these coordinates, if $u(t,x)$ solves then $\theta(\tau,\eta)$ solves, $$\partial_\tau \theta = {\nabla}\cdot (\eta \theta) + \Delta \theta^m - e^{(1 - \alpha - \beta)\beta^{-1}\tau} {\nabla}\cdot(\theta ({e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot) \ast \theta})). \label{eq:asymptotic_pde}$$ Moreover, $\mathcal{U}(t,x;M)$ is stationary in these coordinates, and will be denoted by $\theta_M(\eta)$. That is (see [@CarrilloToscani00]), $$\mathcal{U}(t,x;M) = \left(1 + \frac{t}{\beta}\right)^{-d\beta}\theta_M\left( \left(1 + \frac{t}{\beta}\right)^{-\beta}x\right) = e^{-d\tau}\theta_M(\eta). \label{eq:BarenblattToGS}$$ In fact, $\theta_M(\eta)$ is the unique non-negative solution with mass $M$ to the (degenerate, if $m > 1$) elliptic equation $$0 = {\nabla}\cdot (\eta \theta) + \Delta \theta^m. \label{eq:ground_state}$$ In what follows we will refer to $\theta_M$ as the *ground state Barenblatt solution*. Clearly, ground state solutions are stationary solutions of the homogeneous Fokker-Plank equation $$\partial_\tau \theta = {\nabla}\cdot (\eta \theta) + \Delta \theta^m. \label{def:HFkP}$$ Therefore, the asymptotic convergence to self-similar profiles of solutions to is equivalent to the convergence to the stationary profiles of . This was the fundamental observation made in [@CarrilloToscani00] and is the purpose of the rescaling .

A primary step to proving Theorems \[thm:IA\] and \[thm:IA2\] is establishing that $\theta(\tau,\eta) \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$. By the change of variables, this is precisely the decay estimate stated in Theorem \[thm:Decay\]. This estimate is what allows us to treat the inhomogeneous non-local term in as a vanishing perturbation of . The decay estimate ${\|u(t)\|}_\infty \lesssim t^{-d\beta}$, or equivalently, ${\|\theta(\tau)\|}_\infty