  We study the stability and instability of ALE Ricci-flat metrics around which
a Lojasiewicz inequality is satisfied for a variation of Perelman's
$\lambda$-functional adapted to the ALE situation and denoted
$\lambda_{\operatorname{ALE}}$. This functional was introduced by the authors
in a recent work and it has been proven that it satisfies a good enough
Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat
metrics in dimension larger than or equal to 5.
