  In this paper, we present three exact solutions to the Einstein field
equations, each showing different black hole models. The first solution
introduces a black hole with a variable equation of state, \( P=k(r)\rho \),
that can represent both singular and regular black holes based on parameters \(
M_0 \) and \( w_0 \). The second solution features a black hole with Hagedorn
fluid, relevant for the late stages of black hole formation, and reveals
similarities to the first solution, describing both singular and regular black
holes in a specific case. Furthermore, we investigate the shadow cast by these
black hole solutions to constrain their parameters. Recognizing that real
astrophysical black holes are dynamic, we developed a third, dynamical solution
that addresses gravitational collapse and suggests the potential formation of
naked singularities, indicating that a black hole can transition from regular
to singular and back to regular during its evolution.
