Dynamics of nonlinear scalar field with Robin boundary condition on the Schwarzschild--Anti-de Sitter background

This work concerns the dynamics of conformal cubic scalar field on a Schwarzschild--anti-de Sitter background. The main focus is on understanding how it depends on the size of the black hole and the Robin boundary condition. We identify a critical curve in the parameter space that separates regions with distinct asymptotic behaviours. For defocusing nonlinearity, the global attractor undergoes a pitchfork bifurcation, whereas for the focusing case, we identify a region of the phase space where all solutions blow up in finite time. In the course of this study we observe an interplay between black hole geometry, boundary conditions, and the nonlinear dynamics of scalar fields in asymptotically anti-de Sitter spacetime.