The asymptotic expansion of the solution to the Cauchy problem for a parabolic equation in a perforated space
In this paper the author deals with the Cauchy problem for a second order parabolic equation: \[ {{\partial u_\varepsilon}\over{\partial t}} - {{\partial}\over{\partial x_i}} \bigl( a_{ij}(\varepsilon^{-1}x,\varepsilon^{-2k} t) {{\partial u_\varepsilon}\over{\partial x_j}} \bigr) = f, \] where \(x\) belongs to a perforated domain \({\mathbb R}^n_\varepsilon\) included in \({\mathbb R}^n\), \(t\) belongs to \([0,T]\), and the coefficients \(a_{ij}\) are rapidly oscillating with period depending on \(\varepsilon\). The author constructs a complete asymptotic expansion for the Cauchy problem solution and obtains an estimate for the remainder term in this expansion.
scientific article; zbMATH DE number 1259427