Null controllability of parabolic equations with interior degeneracy and one-sided control.

For $\alpha\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\vert x \vert ^\alpha u_x)_x\qquad (1eigenfunctions of the associated stationary operator in terms of Bessel functions and their zeroes for both weakly and strongly degenerate problems. Hence, we obtain lower $L^2$ bounds for the eigenfunctions on the control region in the case $\alpha \in [0,1)$ and deduce the lack of observability in the case of $\alpha \in [1,2)$. We also provide numerical evidence to illustrate our theoretical results.