Homogenization of parabolic systems with singular perturbations (Q6116029)

The paper under review deals with the study of the convergence rates in periodic homogenization of second-order parabolic systems with fourth-order singular perturbations. More precisely, the authors consider the second-order parabolic system \[ \partial_t u_\varepsilon +\mathcal{L}_\varepsilon u_\varepsilon=F \qquad \text{in } \Omega \times (0,T), \] where \[ \mathcal{L}_\varepsilon=\kappa^2\Delta^2-\operatorname{div}(A(x/\varepsilon,t/\varepsilon^2)\nabla), \qquad 0<\kappa, \varepsilon <1, \] where \(A\) is real, bounded, measurable and satisfies ellipticity and periodicity assumptions. The authors obtain convergence rates depending on \(\kappa\) and \(\varepsilon\) for the above equations with Dirichlet conditions, i.e., \[ u_\varepsilon=0,\qquad \frac{\partial}{\partial \nu_\varepsilon}u_\varepsilon=0, \qquad \text{on } \partial \Omega \times (0,T), \] or Navier conditions, i.e., \[ u_\varepsilon=\Delta u_\varepsilon=0, \qquad \text{on } \partial \Omega \times (0,T). \]