Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpi\'nski gasket

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpi\'nski gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle creation and annihilation with birth-death rates depending locally on the particle configuration. In the macroscopic limit, the particle density evolves according to a nonlinear reaction--diffusion equation, where the reaction term is explicitly determined by the microscopic rates. The key new ingredient is a replacement lemma adapted to the fractal geometry of the Sierpi\'nski gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpi\'nski gasket graph, which require new arguments due to the absence of classical lattice structures.