The Kerr solution is known as the most important model of a rotating black hole (BH) in general relativity (GR). One of the most appealing properties of the maximal analytic extension of the Kerr solution is that, in addition to its event horizon separating the domain of outer communication from the BH, inside the BH region, there exists another "inner-horizon" that is, in fact, a Cauchy horizon. The latter is a null hypersurface beyond which the predictive power of GR is known to be lost. If, for instance, it was possible to cross it by maneuvering suitably, a hypothetical observer could arrive at another epoch or into another universe through a "white hole" horizon. Therefore, the physical properties of the Kerr BH spacetime near its inner horizon are of vital interest, especially from the point of view of Penrose's strong cosmic censor hypothesis. We will study the stability of the inner Cauchy horizons by using analytic and numeric methods in dynamical near Kerr BH spacetimes. 1) Rácz, I., Tóth, G.Zs.: Numerical investigation of the late-time Kerr tails, Class. Quant. Grav. 28 195003 (2011) 2) Csukás, K., Rácz, I., Tóth, G.Zs.: Numerical investigation of the dynamics of linear spin ‘s’ fields on Kerr background I: Late time tails of spin s = ±1, ±2 fields, Phys.Rev.D 100, 104025 (2019) 3) Csukás, K., Rácz, I., : Numerical investigation of the dynamics of linear spin ‘s’ fields on a Kerr background II: Superradiant scattering, Phys. Rev. D 103, 084035 (2021)