Renormalization-group improved Schwarzschild black hole: shadow, ringdown, and strong cosmic censorship

A renormalization-group (RG) improved Schwarzschild-like black hole is investigated, whose lapse function interpolates between the classical Schwarzschild exterior and a quantum-smoothed interior governed by the cutoff scale $\xi$ and interpolation parameter $\gamma$. The horizon structure, photon sphere, and shadow radius $R_{\mathrm{sh}}$ are derived, while scalar, electromagnetic, and Dirac Regge--Wheeler--Zerilli perturbations are treated in a unified framework. Fundamental and overtone quasinormal modes are obtained through sixth-order WKB calculations and checked against time-domain ringdown profiles. Strong Cosmic Censorship (SCC) is tested at the inner Cauchy horizon, generated here without charge or rotation, and the ratio $\beta=|\mathrm{Im}\,\omega|/\kappa_-$ is found to be multipole-independent at the $6\%$ level, following $\beta\simeq\lambda_L/\kappa_-$. Thermodynamic analysis reveals a Davies-type phase transition at the outer horizon and a nontrivial Weinhold--Ruppeiner geometry on the $(S,\gamma)$ slice. The Schwarzschild decay law $T_H\propto 1/r_+$ is replaced by a bell-shaped profile with a maximum $T_H^{\max}\simeq 0.062$. A scan over the $(\xi,\gamma)$ plane captures the joint behavior of the shadow, scalar barrier, SCC ratio, and Hawking temperature, revealing a thin crescent near the extremal boundary where Christodoulou-SCC is marginally violated. Comparison with Bardeen, Hayward, and Bonanno--Reuter black holes shows that the present solution is the most Schwarzschild-like member of the regular-black-hole family at matched perturbation scale, while remaining shadow-degenerate with Hayward and Bonanno--Reuter geometries at the $1\%$ level. Finally, the sparsity of the Hawking flux and the energy-emission rate are analyzed, both depending on $(\xi,\gamma)$ through a single auxiliary function tied to the outer-horizon surface gravity.