Higher-order asymptotic profiles of solutions to the Cauchy problem for the convection-diffusion equation with variable diffusion

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.