Stability for semilinear parabolic problems in $L_2$, $W^{1,2}$, and   interpolation spaces
  An asymptotic stability result for parabolic semilinear problems in $L_2(\Omega)$ and interpolation spaces is shown. Some known results about stability in $W^{1,2}(\Omega)$ are improved for semilinear parabolic mixed boundary value problems. The approach is based on Amann's power extrapolation scales. In a Hilbert space setting, a better understanding of this approach is provided for operators satisfying Kato's square root problem; as a side result some equivalent characterizations of these operators are obtained. 
Pavel Gurevich and Martin V\"ath
math.AP math.DS math.FA