Null Controllability for Degenerate Parabolic Operators

why null controllability fails for α ≥ 2• classical change of variable (Courant-Hilbert)y(x) =∫ 1xdss α/2 U(y(x), t) = x −α/4 u(x, t)transforms equation into˜ω =]˜b, ã[ boundedU t − U yy + V α (y)U = χ eω F0 < y < ∞V α (y) = α 2( 34 α − 1) 1[2−(2−α)y] 2 bounded for α ≥ 2• Escauriaza, Seregin, Sverak (2003, 2004)U(·, T ) = 0 ⇐⇒ spt ( U(·, 0) ) ⊂ [0, ã[• u t − ( x α u x)x = χ ωf{no [0, 1]yes if spt(u 0 ) ⊂]a, 1]