Stabilization via Homogenization

TL;DR

This paper studies a 1+1-dimensional system with regions of hyperbolic and elliptic types, showing that as the system's heterogeneity increases, solutions converge to an exponentially stable limit equation, highlighting the stabilizing effect of homogenization.

Contribution

It demonstrates that a system with mixed hyperbolic and elliptic regions converges to an exponentially stable limit equation through homogenization.

Findings

Solutions converge weakly to a stable limit equation

Homogenization induces exponential stability in the limit

Replacing elliptic with parabolic loses exponential stability

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 $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{j-1}{n},\frac{2j-1}{2n}\big)$ and $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{2j-1}{2n},\frac{j}{n}\big)$. 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.