Stabilization via Homogenization

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, ${\partial }_t^2{u}_n-{\partial }_x^2{u}_n={\partial }_{t}f$ and $u_n-{\partial }_x^2{u}_n=f$ on the respective spatial domains ${\bigcup }_{j\in \{1,\dots ,n\}}(\frac{j-1}{n},\frac{2j-1}{2n})$ and ${\bigcup }_{j\in \{1,\dots ,n\}}(\frac{2j-1}{2n},\frac{j}{n})$. We show that $(u_n{)}_n$ converges weakly to $u$, which solves the exponentially stable limit equation ${\partial }_t^{2}u+2{\partial }_{t}u+u-4{\partial }_x^{2}u=2(f+{\partial }_{t}f)$ on $[0,1]$. If the elliptic equation is replaced by a parabolic one, the limit equation is \emph{not} exponentially stable.

8 pages; some typos corrected; referee's comments incorporated