Interacting k-essence field with non-pressureless Dark Matter: Cosmological Dynamics and Observational Constraints

We investigate a class of interacting dark energy and dark matter (DM) models, where dark energy is modeled as a $k$-essence scalar field with an inverse-square potential. Two general forms of interaction are considered: one proportional to the Hubble parameter, and another independent of the Hubble parameter, depending instead on combinations of the energy densities and pressures of the dark sectors. {The cosmological evolution is reformulated in terms of an autonomous system of equations, which provides a convenient phase-space parametrization for the numerical integration of the background dynamics and for confronting the models with observations.} The models are tested against a wide range of observational datasets, including cosmic chronometers (CC), BAO measurements from DESI DR2, compressed Planck data (PLA), Pantheon+ (PP), DES supernovae, Big Bang Nucleosynthesis (BBN), and strong lensing data from H0LiCOW (HCW). The analysis shows that the models consistently reproduce all major cosmological epochs and yield statistically competitive results compared to the flat $\Lambda$CDM model. The models exhibit late-time de-Sitter solutions, ensuring ghost-free evolution, with the Hubble constant in the range $H_0 \sim 67$--$70$ km/s/Mpc.