  The nature of the progenitors and explosion mechanism of Type Iax supernovae
(SNe Iax) remain a mystery. The single-degenerate (SD) systems that involve the
incomplete pure deflagration explosions of near-Chandrasekhar-mass white dwarfs
(WDs) have recently been proposed for producing SNe Iax, in which
non-degenerate companions are expected to survive from SN explosions. In this
work we concentrate on the main-sequence (MS) donor SD progenitor systems. By
mapping the computed companion models from three-dimensional hydrodynamical
simulations of ejecta-companion interaction into a one-dimensional stellar
evolution code MESA, we investigate the long-term appearance and observational
signatures of surviving MS companions of SNe Iax by tracing their post-impact
evolution. Depending on different MS companion models, it is found that the
shocked surviving companion stars can significantly expand and evolve to be
more luminous (5-500 Lsun) for a time-scale of 10-1e4 yr. Comparing with the
late-time light curve of an observed SN Iax (SN 2005hk), it is suggested that
surviving MS companions of SNe Iax would expect to be visible about 1000 days
after the explosion when SN itself has been faded.
