Evaporation of dynamical horizon with the Hawking temparature in the {\bf K-}essence emergent Vaidya spacetime

In the {\bf K-}essence Vaidya Schwarzschild spacetime, we apply the dynamical horizon equation to measure the mass-loss due to Hawking radiation and the tunneling formalism (Hamilton-Jacobi method) to calculate the hawking temperature. Assuming the Dirac-Born-Infeld kind of non-standard action for the {\bf K-}essence here, the background physical spacetime is a static spherically symmetric black hole, and we constrain the {\bf K-}essence scalar field to be a function only of either forward or backward time. The {\bf K-}essence emergent gravity and the generalizations of Vaidya spacetime have been linked by Manna et al. In this paper, we use Sawayama's modified description of the dynamical horizon to show that the obtained findings deviate from the standard Vaidya spacetime geometry.