Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $u(t) - (a(x) u(x))(x) + c(t, x) u = h(t, x), (t, x) is an element of (0, T) x (0, 1),$ where $a(.)$ is degenerate at $0$. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of $a(.)$. Then, we study the null controllability on $[0, 1]$ of the semilinear degenerate parabolic equation $u(t) - (a( x) u(x))(x) + f (t, x, u) = h(t, x)chi(omega)( x),$ where $( t, x)$ is an element of $(0, T) x (0, 1), omega = (alpha, beta)$ subset of $[0, 1]$, and $f$ is locally Lipschitz with respect to $u$.