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, ∂2 t un − ∂2 xun = ∂tf and un−∂ xun = f on the respective spatial domains ⋃ j∈{1,...,n} ( j−1 n , 2j−1 2n ) and ⋃ j∈{1,...,n} (2j−1 2n , j n ) . We show that (un)n converges weakly to u, which solves the exponentially stable limit equation ∂2 t u+ 2∂tu+ u− 4∂2 xu = 2(f + ∂tf) on [0, 1]. If the elliptic equation is replaced by a parabolic one, the limit equation is not exponentially stable.