The following problem is from a previous qualification exam that I am stuck on. I only need help with part $c)$ as $a)$ and $b)$ are just computation.
Consider $\rho(x,t)$ solving the nonlinear diffusion equation with drift:
$$\rho_t - \Delta(\rho^2) - \nabla \cdot(x|x|^2 \rho) = 0 \quad \text{ in } (x,t) \in \mathbb{R}^2 \times \{t>0\}
$$
where the initial data $\rho_0(x) \geq 0$ is compactly supported and $\int_{\mathbb{R}^2} \rho_0 = 1$ . Let us assume that $\rho$ stays compactly supported and non-negative for all times $t > 0$. I have already solved $a)$ and $b)$, and I am stuck on $c)$.
Part $a)$ of the question was show for any time $t >0$ $$ \int\limits_{\mathbb{R}^2} \rho(x,t) \mathrm{d}x \mathrm{d} t= 1 $$ Part $b)$ Was to show that the following energy is non-increasing in time for any constant $C$ $$ E(t) := \int\limits_{\mathbb{R}^2} \left[\rho^2(x,t) +\rho(x,t)\frac{|x|^4}{4} + C \rho(x,t)\right] \mathrm{d}x \mathrm{d}t $$ Part $c)$ Use $(a)-(b)$ to show that $\rho(\cdot,t)$ converges as $t \rightarrow \infty$ to the stationary profile $(C_0 - \frac{|x|^4}{4})_+$ for an appropriate $C_0$. $(f)_+ := \max(f,0)$. (the positive part of $f$)
My Attempt. We know that $\frac{\partial E}{\partial t} \leq 0$, so
$$
\int_{\mathbb{R}^2} \left[ 2\rho(x,t)\rho_t(x,t) + \rho_t(x,t)\frac{|x|^4}{4}\right]\mathrm{d}x \mathrm{d}t\leq 0
$$
and that
$$
\int_{\mathbb{R}^2} \rho_t(x,t) \mathrm{d}x \mathrm{d}t = 0
$$
I want to some how use $\partial_t E \leq 0$ to show that $\rho_t \leq 0$ as $t \rightarrow \infty$ to get $\rho_t \rightarrow 0$ as $t \rightarrow \infty$, so that we get the solution converges to the solution of
$$
\Delta(\rho^2) + \nabla \cdot (x|x|^2\rho) = 0
$$
which I assume $(C - \frac{|x|^4}{4})$ is a solution for when its positive and the constant $C_0$ should be chosen so that it gives $(C-|x|^4/4)_+$ unit mass. Another thing I noticed is that the energy with $C = -C_0$ is
$$
E(t) = \int_{\mathbb{R}^2} \rho(x,t) \left[\rho(x,t) + \frac{|x|^4}{4} - C_0\right] \mathrm{d}x \mathrm{d}t
$$
is very similar to $|\rho(x,t) - (C_0 - \frac{|x|^4}{4})_+|$, but I am unsure how to obtain just the positive part into the formulation of the energy and how to control the evolution of the compact supports (what stops the compact support to evolving to the empty set as. $t \rightarrow \infty$?) I am also confused on how to even use $t \rightarrow \infty$ on either energy to get control of $\rho$. Any hints or advice would be really appreciated!