Stabilization via homogenization. (October 2016)

• Record Type: Journal Article
• Authors: Waurick, Marcus
• Is Part Of: Applied mathematics letters. Volume 60(2016:Oct.)
• Page Start: 101
• Page End: 107
• Publication Date: 2016-10
• Subjects: Evolutionary equations -- Equations of mixed type -- Homogenization -- Exponential stability; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2016.04.004
• Languages: English
• ISSNs: 0893-9659

Abstract

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.