Neumann boundary condition for strong degenerate problems

Question

In this article 1, in (1.4), the authors say tha the natural boundary condition to the strong degenerate problem is of Neumann type. I couldn't find the reason. Can someone explain the reason or give me some reference?

Answer 1

I think the best answer is given in the book P. Cannarsa et al. Global Carleman Estimates for Degenerate Parabolic Operators.

The degeneracy of the problem $$ u_t - (a(x) u_x)_x = h \chi_\omega $$ depends on $a(x)$.

If $\frac{1}{a} \in L^1(0, 1)$ we have weakly degenerate operators, if $\frac{1}{a} \notin L^1(0, 1)$ we have strongly degenerate operators.

For the weakly degenerate operators one can consider Dirihlet and Neumann boundary conditions. For the strongly degenerate operators the boundary conditions is $$ \begin{cases} a(x)u_x(t, x) \to 0 \mbox{ as } x \to 0^+ \\ u(t, 1) = 0 \end{cases} $$

This boundary conditions is of Neumann type because we use $u_x(t, x)$. It was proved that the problem is well-posed with given above boundary conditions and the solution belongs to suitable weighted Sobolev space.