  We consider the asymptotic behavior of solutions to the convection-diffusion
equation: \[ \partial_t u - \mathrm{div}\left(a(x)\nabla u\right) =
d\cdot\nabla \left(\left\lvert u\right\rvert ^{q-1}u\right),\ \
x\in\mathbb{R}^n, \ t>0 \] with an integrable initial data $u_{0}(x)$, where
$n\ge1$, $q>1+\frac{1}{n}$ and $d\in \mathbb{R}^{n}$. Moreover, we take
$a(x)=1+b(x)>0$, where $b(x)$ is smooth and decays fast enough at spatial
infinity. It is known that the asymptotic profile of the solution to this
problem can be given by the heat kernel. Moreover, some higher-order asymptotic
expansions of the solution have already been studied. In particular, the
structures of the second asymptotic profiles strongly depend on the nonlinear
exponent $q$. More precisely, these profiles have different decay orders in
each of the following three cases: $1+\frac{1}{n}<q<1+\frac{2}{n}$;
$q=1+\frac{2}{n}$; $q>1+\frac{2}{n}$. In this paper, we focus on the critical
case $q=1+\frac{2}{n}$. By analyzing the corresponding integral equation in
details, we have succeeded to give the more higher-order asymptotic expansion
of the solution, which generalizes the previous works.
