Thermodynamics, phase structure and quasinormal modes for AdS Heyward massive black hole

In this paper, we find the new exact AdS black hole (BH) solution in the presence of massive gravity and nonlinear electrodynamics (NED). The obtained black hole solution (known as AdS Heyward massive BH ) interpolates with the AdS Heyward BH in the absence of graviton mass (m) and massive BH when the magnetic charge is switched off. We calculate the exact expression of thermodynamics quantities including local (heat capacity), global (free energy), and dynamical stability (quasinormal modes) of the obtained BH solution. The heat capacity of the BH diverges where the temperature is maximum and free energy is minimum. We also study the extended thermodynamics of the BH when the cosmological constant (Lambda) is treated as the thermodynamics pressure (P = -Lambda/8 pi). We analyze the first and second-order phase transition by studying the behavior of Gibbs free energy and these phase transitions are similar to the van der Walls phase transition. The effect of magnetic charge and graviton mass are opposite to each other on critical values.