The Singular Diffusion Equation with Boundary Degeneracy

Abstract: For the heat conduction on a bounded domain with boundary degeneracy, though its diffusion coefficient vanishes on the boundary, it is still possible that the heat flux may transfer across the boundary. A known result shows that the key role is the ratio of the diffusion coefficient near the boundary. If this ratio is large enough, the heat flux transference has not any relation to the boundary condition but is completely controlled by the initial value. This phenomena shows there are some essential differences between the heat flux with boundary degeneracy and that without boundary degeneracy. However, under the assumption on the uniqueness of the weak solutions, the paper obtains that the weak solution of the singular diffusion equation with boundary degeneracy, has the same regular properties as the solution of a singular diffusion equation without boundary degeneracy.