The recent detection of gravitational waves and electromagnetic counterparts from the double neutron star merger event GW+EM170817, supports the standard paradigm of short gamma-ray bursts (SGRBs) and kilonovae/macronovae. It is important to reveal the nature of the compact remnant left after the merger, either a black hole or neutron star, and their physical link to the origin of the long-lasting emission observed in SGRBs. The diversity of the merger remnants may also lead to different kinds of transients that can be detected in future. Here we study the high-energy emission from the long-lasting central engine left after the coalescence, under certain assumptions. In particular, we consider the X-ray emission from a remnant disk and the non-thermal nebular emission from disk-driven outflows or pulsar winds. We demonstrate that late-time X-ray and high-frequency radio emission can provide useful constraints on properties of the hidden compact remnants and their connections to long-lasting SGRB emission, and we discuss the detectability of nearby merger events through late-time observations at $sim30-100$ d after the coalescence. We also investigate the GeV-TeV gamma-ray emission that occurs in the presence of long-lasting central engines, and show the importance of external inverse-Compton radiation due to up-scattering of X-ray photons by relativistic electrons in the jet. We also search for high-energy gamma-rays from GW170817 in the Fermi-LAT data, and report upper limits on such long-lasting emission. Finally, we consider the implications of GW+EM170817 and discuss the constraints placed by X-ray and high-frequency radio observations.
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