Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging

V. V. Zhikov ; S. E. Pastukhova

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 223-246

Résumé

The diffusion equation is considered in an infinite $1$-periodic medium. For its fundamental solution we find approximations at large values of time $t$. Precision of approximations has pointwise and integral estimates of orders $O(t^{-\frac{d+j+1}2})$ and $O(t^{-\frac{j+1}2}),$ $j=0,1,\dots$, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in $L^p$-spaces, $1\le p\le\infty$. For the analogous equation in an $\varepsilon$-periodic medium (here $\varepsilon$ is a small parameter) we obtain approximations of the operator exponential in $L^p$-operator norms for a fixed time with precision of order $O(\varepsilon^n)$, $n=1,2,\dots$.