 $M = {\|u_0\|}_1$ given in or .

\[thm:IA2\] Let $d \geq 2$ and ${\mathcal{K}}$ be admissible with ${\nabla}{\mathcal{K}}(x) = \mathcal{O}({\left\vertx\right\vert}^{-\gamma})$ as ${\left\vertx\right\vert} \rightarrow \infty$ for some $\gamma \in [d-1,d]$. If $\gamma = d-1$ then suppose $m  \in [1,2-2/d)$ and otherwise we may take $m \in [1,2-2/d]$. Suppose $u(t)$ is a global weak solution of which satisfies the decay estimate . Then, for all $\delta>0$, $u(t)$ satisfies $$t^{d\beta\left(1 - \frac{1}{p}\right)}{\|u(t) - \mathcal{U}(t;M)\|}_p \lesssim_{\delta} (1+t)^{-\frac{\beta}{p}\min\left(1, 1+\gamma - \beta^{-1} - \delta \right)}, \;\; \forall p, \, 1 \leq p < \infty. \label{ineq:conv_IA2}$$ Here $\beta$ and $\mathcal{U}(t,x;M)$ are as above.

Note that $u_0 \in L_+^1({\mathbb R}^d; (1 + {\left\vertx\right\vert}^2)dx )\cap L^\infty({\mathbb R}^d)$ implies $u_0 \log u_0 \in L^1({\mathbb R}^d)$ by Jensen’s inequality.

The convergence rate in is optimal, as it matches that of the corresponding diffusion equation. Optimality is not known for or , however we suspect that these rates are nearly optimal. Note that the convergence rate obtained in reduces to and when $\gamma = d$. Moreover, if $\gamma = d-1$, then the convergence rate goes to zero as $m \nearrow 2 -2/d$.

In some cases, it may be possible to show that the size conditions of *(i)* of Theorem \[thm:Decay\] are only required of the $L^{(2-m)d/2}$ norm, which is the norm left invariant by the scaling symmetry of when ${\mathcal{K}}$ is the Newtonian potential. Indeed, one can prove global existence and uniform boundedness of solutions to in these supercritical regimes using iteration techniques without any smallness assumption on the mass [@Corrias04; @SugiyamaDIE06; @SugiyamaADE07]. However, in [@LionsCC84] it is shown that depending on the singularity of the kernel, if the mass is sufficiently large, there may exist stationary solutions for all $m$, $1 < m < 2$. Since these are potentially included in our analysis, in order to state decay results in