Averaging for fundamental solutions of parabolic equations
A parabolic operator of order \(2p\) depending on the small parameter \(\varepsilon\) \[ \partial_t-\sum_{q\leq 2p}A_q(x,t/\varepsilon)\partial^q_x \] is considered in \(d\)-dimensional \(x\)-space. The coefficients \(A_q\) depend on the fast time under basic hypothesis that there exists an averaging: \[ \int^t_0\bigl[A_q(x,s/\varepsilon)-A^0_q(x)\bigr]ds\to 0\quad\text{as }\varepsilon\to 0. \] The corresponding parabolic operator with the coefficients \(A^0_q\) is taken and two fundamental solutions \(\Gamma_\varepsilon\), \(\Gamma_0\) are considered. The main goal of the paper is justification of the convergence \(\partial^j_x(\Gamma_\varepsilon-\Gamma_0)\to 0\) as \(\varepsilon\to 0\), \((0\leq j\leq 2p-1)\).
scientific article; zbMATH DE number 1014431
scientific article