It is expected that a pressure bump can be formed at the inner edge of a dead-zone, and where vortices can develop through the Rossby Wave Instability (RWI). It has been suggested that self-gravity can significantly affect the evolution of such vortices. We present the results of 2D hydrodynamical simulations of the evolution of vortices forming at a pressure bump in self-gravitating discs with Toomre parameter in the range $4-30$. We consider isothermal plus non-isothermal disc models that employ either the classical $beta$ prescription or a more realistic treatment for cooling. The main aim is to investigate whether the condensating effect of self-gravity can stabilize vortices in sufficiently massive discs. We confirm that in isothermal disc models with ${cal Q} gtrsim 15$, vortex decay occurs due to the vortex self-gravitational torque. For discs with $3lesssim {cal Q} lesssim 7$, the vortex develops gravitational instabilities within its core and undergoes gravitational collapse, whereas more massive discs give rise to the formation of global eccentric modes. In non-isothermal discs with $beta$ cooling, the vortex maintains a turbulent core prior to undergoing gravitational collapse for $beta lesssim 0.1$, whereas it decays if $beta ge 1$. In models that incorpore both self-gravity and a better treatment for cooling, however, a stable vortex is formed with aspect ratio $chi sim 3-4$. Our results indicate that self-gravity significantly impacts the evolution of vortices forming in protoplanetary discs, although the thermodynamical structure of the vortex is equally important for determining its long-term dynamics.
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