Homogenization of fundamental solutions for parabolic operators involving non-self-similar scales

We establish the asymptotic expansion of the fundamental solutions with precise error estimates for second-order parabolic operators \(\begin{aligned} \partial _t -\text {div}(A(x/\varepsilon , t/\varepsilon ^\ell )\nabla ), \quad \, 0<\varepsilon<1,\, 0<\ell <\infty ,\end{aligned}\) $$\begin{array}{c}\hfill {\partial }_t-\text{div}(A(x/\epsilon ,t/{\epsilon }^{\ell })\mathrm{\nabla }),0<\epsilon <1,0<\ell <\infty ,\end{array}$$ in the case \(\ell \ne 2,\) $\ell \ne 2,$ where the spatial and temporal variables oscillate on non-self-similar scales and do not homogenize simultaneously. To achieve the goal, we explore the direct quantitative two-scale expansions for the aforementioned operators, which should be of some independent interests in quantitative homogenization of parabolic operators involving multiple scales. In the self-similar case \(\ell =2\) $\ell =2$ , similar results have been obtained in Geng and Shen (Anal PDE 13(1): 147–170, 2020).