On a general degenerate/singular parabolic equation with a nonlocal space term

In this paper, we study the null controllability for the problems associated to the operators y t - A y - λ b ( x ) y + ∫ 0 1 K ( t ,x ,τ ) y ( t ,τ ) d τ ,( t ,x ) ∈ ( 0 ,T ) × ( 0,1) where A y : = a y x x or A y : = ( a y x ) x and the functions a and b degenerate at an interior point x 0 ∈ ( 0,1 ) . To this aim, as a first step, we study the well-posedness, the Carleman estimates, and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then, using Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.