A linear homogenization problem with time dependent coefficient

Abstract:

We consider: the homogenization problem \[ \begin {cases} (\partial u\varepsilon /\partial t)(x,t) + \beta _\varepsilon (x) u_\varepsilon (x,t) = 0, & t\leqslant 0, \\ u_\varepsilon (x,0) = \phi (x), \end {cases} \] where $\beta$ is a strictly positive bounded real function, periodic of period $1$, and ${\beta _\varepsilon }(x) = \beta (x/\varepsilon )$; the equivalent integral equation \[ {u_\varepsilon }(x,t) + \int _0^t {{\beta _\varepsilon }(x) {u_\varepsilon }(x,s)\;ds = \phi (x)}; \] and the homogenized equation \[ {u_0}(x,t) + \int _0^t {K(t - s) {u_0}(s) ds = \phi (x)}, \] where $K$ is a unique, well-defined function depending on $\beta$. We study this problem for a time dependent $\beta$, and characterize a two-variable function $K(s,t)$ satisfying \[ {u_0}(x,t) + \int _0^t {K(s,t - s) {u_0}(x,s)\;ds = \phi (x)} \] and study its uniqueness.