The existence of periodic solutions to doubly degenerate Allen-Cahn equation with Neumann boundary condition

This work is concerned with the periodic solution of a doubly degenerate Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Firstly, we define a new associated auxiliary problem. Secondly, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, where this limit point represents a nontrivial nonnegative time-periodic solution of the main studied problem. It is observed that the topological degree theorem technique plays an important role in proving the desired results. Furthermore, this technique can be applied to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.