Null Controllability for Degenerate Parabolic Operators

well-posednessin both cases{D(A) ={u ∈ H1a (0, 1) ∣ ∣ au x ∈ H 1 (0, 1)}Au = ( au x)xgenerates analytic semigroup in L 2 (0, 1)(u• t − `a(x)u x´= f in Q xT =]0, 1[×]0, T [u(x, 0) = u 0 (x)unique solution u ∈ C(0, T ; L 2 (0, 1)) ∩ L 2 (0, T ; Ha(0, 1 1))• maximal regularityu 0 ∈ H 1 a(0, 1) =⇒ u ∈ H 1 (0, T ; L 2 (0, 1))∩L 2 (0, T ; D(A))• strongly degenerate caseu ∈ D(A) =⇒ au x(x→0)−→ 0