  The inclusion of Localized sources of matter (LSM) in the energy-momentum
tensor has been widely utilized in the construction of regular black holes
(RBHs). In this work, we provide a recipe for generating new 4D and
extra--dimensional non regular black hole solutions based on the incorporation
of LSM. Our proposal, instead of forming a central de Sitter core (as observed
in RBHs), leads to the formation of a central integrable singularity, without
the presence of an inner horizon (as occurs in RBSs). Based on this recipe, we
also present a new 4D and extra-dimensional model (new metric and new energy
density) of this type of black hole.
  Furthermore, it is known that introducing sources of matter into RBHs leads
to the formation of a black remnant once the evaporation is completed,
preventing complete evaporation down to $r_h=0$. We demonstrate that by
introducing our proposed type of matter, the absence of an inner horizon allows
for complete evaporation down to $r_h=0$ without the formation of a remnant,
both in the $4D$ and the extra-dimensional cases. The complete evaporation of
our model also differs from the Schwarzschild vacuum solution, which requires
an infinite temperature to reach complete evaporation down to $r_h=0$ in the
evaporation process. In our model, complete evaporation is achieved at a finite
temperature for the 4D case and at zero temperature for the extra-dimensional
case. Our type of LSM induces the appearance of a term of work done by the
system on the external environment, denoted as $dW$, in the first law of
thermodynamics, i.e. $dU=dM=TdS-dW$, which reduces the usual internal energy
change $dU=dM=TdS$. Furthermore, a linear correction to the usual area law of
entropy emerges, which differs from the corrections generated by matter in
RBHs. In $4D$ scenarios, our entropy correction could be associated with
quantum effects through GUP corrections.

General Relativity and Quantum Cosmology