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In this work we construct families of anisotropic neutron stars for anequation of state compatible with the constraints of the gravitational-waveevent GW170817 and for four anisotropy ansatze. Such stars are subjected to aradial perturbation in order to study their stability against radialoscillations and we develop a dynamical model to describe the non-adiabaticgravitational collapse of the unstable anisotropic configurations whoseultimate fate is the formation of a black hole. We find that the standardcriterion for radial stability $dM/d\rho_c >0$ is not always compatible withthe calculation of the oscillation frequencies for some anisotropy ansatze, andeach anisotropy parameter is constrained taking into account the recentrestriction of maximum mass of neutron stars. We further generalize the TOVequations within a non-adiabatic context and we investigate the dynamicalbehaviour of the equation of state, heat flux, anisotropy factor and massfunction as an unstable anisotropic star collapses. After obtaining theevolution equations we recover, as a static limit, the background equations.

Equilibrium, radial stability and non-adiabatic gravitational collapse of anisotropic neutron stars