Stabilization via homogenization. (October 2016) - Waurick, Marcus
Abstract: 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, ∂ t 2 u n − ∂ x 2 u n = ∂ t f and u n − ∂ x 2 u n = f on the respective spatial domains ⋃ j ∈ 1, …, n ( j − 1 n, 2 j − 1 2 n ) and ⋃ j ∈ 1, …, n ( 2 j − 1 2 n, j n ) . We show that ( u n ) n converges weakly to u, which solves the exponentially stable limit equation ∂ t 2 u + 2 ∂ t u + u − 4 ∂ x 2 u = 2 ( f + ∂ t f ) on [ 0, 1 ] . If the elliptic equation is replaced by a parabolic one, the limit equation is not exponentially stable.