Nonlinear parabolic equation with a dynamical boundary condition of diffusive type

We consider the degenerate nonlinear parabolic problem with a dynamical boundary condition∂tθ(u)-Δu=finΩ×(0,T),∂tβ(u)-ΔΓu+u+∇u·ν=0on∂Ω×(0,T),where ΔΓΔΓ stands for Laplace–Beltrami operator. Rothe’s method is used to address the unique solvability of the problem. Error estimates for a time and space discretization are provided. We support theoretical results with numerical experiments.