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We investigate the thermodynamics and the classical and semiclassical
dynamics of two-dimensional ($2\text{D}$), asymptotically flat, nonsingular
dilatonic black holes. They are characterized by a de Sitter core, allowing for
the smearing of the classical singularity, and by the presence of two horizons
with a related extremal configuration. For concreteness, we focus on a
$2\text{D}$ version of the Hayward black hole. We find a second order
thermodynamic phase transition, separating large unstable black holes from
stable configurations close to extremality. We first describe the black-hole
evaporation process using a quasistatic approximation and we show that it ends
in the extremal configuration in an infinite amount of time. We go beyond the
quasistatic approximation by numerically integrating the field equations for
$2\text{D}$ dilaton gravity coupled to $N$ massless scalar fields, describing
the radiation. We find that the inclusion of large backreaction effects ($N \gg
1$) allows for an end-point extremal configuration after a finite evaporation
time. Finally, we evaluate the entanglement entropy (EE) of the radiation in
the quasistatic approximation and construct the relative Page curve. We find
that the EE initially grows, reaches a maximum and then goes down towards zero,
in agreement with previous results in the literature. Despite the breakdown of
the semiclassical approximation prevents the description of the evaporation
process near extremality, we have a clear indication that the end point of the
evaporation is a regular, extremal state with vanishing EE of the radiation.
This means that the nonunitary evolution, which commonly characterizes the
evaporation of singular black holes, could be traced back to the presence of
the singularity.