  In the literature on singular perturbation (Lavrentiev regularization) for
the stable approximate solution of operator equations with monotone operators
in the Hilbert space the phenomena of conditional stability and local
well-posedness and ill-posedness are rarely investigated. Our goal is to
present some studies which try to bridge this gap. So we discuss the impact of
conditional stability on error estimates and convergence rates for the
Lavrentiev regularization and distinguish for linear problems well-posedness
and ill-posedness in a specific manner motivated by a saturation result. The
role of the regularization error in the noise-free case, called bias, is a
crucial point in the paper for nonlinear and linear problems. In particular,
for linear operator equations general convergence rates, including logarithmic
rates, are derived by means of the method of approximate source conditions.
This allows us to extend well-known convergence rates results for the
Lavrentiev regularization that were based on general source conditions to the
case of non-selfadjoint linear monotone forward operators for which general
source conditions fail. Examples presenting the self-adjoint multiplication
operator as well as the non-selfadjoint fractional integral operator and
Ces\`aro operator illustrate the theoretical results. Extensions to the
nonlinear case under specific conditions on the nonlinearity structure complete
the paper.
