Dynamical spontaneous scalarization in Einstein-Maxwell-scalar theory *

We study the linear instability and nonlinear dynamical evolution of Reissner-Nordstr\"om (RN) black hole in Einstein-Maxwell-scalar theory asymptotic flat spacetime. focus on coupling function $f(\phi)=e^{-b\phi^2}$ which allows both scalar-free RN solution scalarized solution. first present system parameters during dynamic scalarization. For parameter regions where spontaneous scalarization occurs, we find that scalar field at horizon is dominated by fundamental unstable mode from analysis early times. At late times, can be viewed as perturbation holes.