Cosmic rays are confined in the atmospheres of galaxy clusters and, therefore, they can play a crucial role in the heating of their cool cores. We discuss here the thermal and non-thermal features of a model of cosmic ray heating of cluster cores that can provide a solution to the cooling-flow problems. To this aim, we generalize a model originally proposed by Colafrancesco, Dar & DeRujula (
2004A&A...413..441C) and we show that our model predicts specific correlations between the thermal and non-thermal properties of galaxy clusters and enables various observational tests. The model reproduces the observed temperature distribution in clusters by using an energy balance condition in which the X-ray energy emitted by clusters is supplied, in a quasi-steady state, by the hadronic cosmic rays, which act as ``warming rays'' (WRs). The temperature profile of the intracluster (IC) gas is strictly correlated with the pressure distribution of the WRs and, consequently, with the non-thermal emission (radio, hard X-ray and gamma-ray) induced by the interaction of the WRs with the IC gas and the IC magnetic field. The temperature distribution of the IC gas in both cool-core and non cool-core clusters is successfully predicted from the measured IC plasma density distribution. Under this contraint, the WR model is also able to reproduce the thermal and non-thermal pressure distribution in clusters, as well as their radial entropy distribution, as shown by the analysis of three clusters studied in detail: Perseus, A2199 and Hydra. The WR model provides other observable features of galaxy clusters: a correlation of the pressure ratio (WRs to thermal IC gas) with the inner cluster temperature (P
WR/P
th)~(kT
inner)
–2/3, a correlation of the gamma-ray luminosity with the inner cluster temperature L
γ~(kT
inner)
4/3, a substantial number of cool-core clusters observable with the GLAST-LAT experiment, a surface brightness of radio halos in cool-core clusters that recovers the observed one, a hard X-ray ICS emission from cool-core clusters that is systematically lower than the observed limits and yet observable with the next generation high-sensitivity and spatial resolution HXR experiments like Simbol-X. The specific theoretical properties and the multi-frequency distribution of the e.m. signals predicted in the WR model render it quite different from the other models so far proposed for the heating of clusters' cool-cores. Such differences make it possible to prove or disprove our model as an explanation for the cooling-flow problems on the basis of multi-frequency observations of galaxy clusters.