Stabilization via homogenization

In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, ∂t2un−∂x2un=∂tfand un−∂x2un=f on the respective spatial domains ⋃j∈{1,…,n}(j−1n,2j−12n) and ⋃j∈{1,…,n}(2j−12n,jn). We show that (un)n converges weakly to u, which solves the exponentially stable limit equation ∂t2u+2∂tu+u−4∂x2u=2(f+∂tf) on [0,1]. If the elliptic equation is replaced by a parabolic one, the limit equation is not exponentially stable.