We investigate quantitative estimates in homogenization of the locally periodic parabolic operator with multiscales \( {\partial _t} - \mathrm{{div}}\left( {A\left( {x,t,x/\varepsilon ,t/{\kappa ^2}} \right) \nabla } \right) ,\;\varepsilon> 0,{\hspace{0.55542pt}} \kappa > 0. \) Under proper assumptions, we establish the full-scale interior and boundary Lipschitz estimates. These results are new even for the case \(\kappa =\varepsilon \) , and for the periodic operators \( {\partial _t} - \mathrm{{div}}\left( {A\left( {x/\varepsilon ,t/{\varepsilon ^\ell }} \right) \nabla } \right) ,\;0< \varepsilon ,\ell < \infty , \) of which the large-scale Lipschitz estimate down to \(\varepsilon +\varepsilon ^{\ell /2}\) was recently established by the Geng and Shen (Arch Ration Mech Anal 236:145–188, 2020). Due to the non-self-similar structure, the full-scale estimates do not follow directly from the large-scale estimates and the blow-up argument. As a byproduct, we also derive the convergence rates for the corresponding initial-Dirichlet problems, which extend the results in the aforementioned literature to more general settings.
