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The study of the nuclear matter equation-of-state (EoS) is a relevant topic of modern nuclear physics. It governs the behaviour of nuclear matter away from the normal conditions found in nuclei interiors and plays a major role in heavy-ion collisions, in determining neutron skin thicknesses of neutron rich nuclei and the mass-radius relation of neutron stars, and in modelizations of supernovae explosions. Its uncertain knowledge is related to difficulties in solving the many-body problem with realistic nuclear interactions. In the last decades several studies, from both theoretical and experimental sides, have allowed relevant progress in the description of the EoS, both for the isospin-symmetric matter and for the isospin-asymmetric one, the so-called symmetry energy, especially at densities below the saturation point. In this paper we review some of the studies on the high-density behavior of the EoS, obtained by studying heavy-ion collisions with incident energies between several hundred MeV up to about 2 GeV per nucleon, with a focus on those carried out at the GSI laboratory in Darmstadt (Germany) using the SIS18 accelerator beams. Constraints on the isospin-symmetric matter EoS, based on studies of kaon and pion production and collective flows, are reviewed. Regarding the symmetry energy, results based on charged pion ratios and neutron-to-charged particles elliptic flow ratios are discussed. A brief overview on the heavy-ion collisions studies sensitive to densities below the nuclear saturation density is presented, but the main emphasis of the review is on the density dependence of the symmetry energy above saturation density, which is also the region which is particularly important for astrophysics. Estimates of neutron star radii, as deduced by the results of heavy-ion collisions discussed in this review, are compared to astrophysics results, including the recent ones made possible by gravitational waves detection and X-ray satellite based observation. A multiple source analysis, using theoretical calculations, astrophysics and heavy-ion collision results, in constraining neutron star radii, is discussed as an illustrative example of the role played by heavy-ion collision results in the multi-messenger astronomy era. Finally, some future perspectives and experimental possibilities are outlined. [ABSTRACT FROM AUTHOR]

Excitation functions of flow and stopping observables for the Au+ Au system at energies from 40 to 1500MeV per nucleon are presented. The systematics were obtained by merging the results of the INDRA and FOPI experiments, both performed at the GSI facility. The connection to the nuclear equation of state is discussed. [ABSTRACT FROM AUTHOR]

Directed and elliptic flow for the 197Au+197Au system at incident energies between 40 and 150 MeV per nucleon has been measured using the INDRA 4π multi-detector. For semi-central collisions, the excitation function of elliptic flow shows a transition from in-plane to out-of-plane emission at around 100 MeV per nucleon. The directed flow changes sign at a bombarding energy between 50 and 60 MeV per nucleon and remains negative at lower energies. Molecular dynamics calculations (CHIMERA) indicate sensitivity of the global squeeze-out transition on the σ NN and demonstrate the importance of angular momentum conservation in transport codes at low energies. [ABSTRACT FROM AUTHOR]