Null controllability properties of some degenerate parabolic equations.

But the Hardy type inequalities allow (once again) to correct this
when α < 2, and to obtain in particular
∫∫
(2 − α) ρ 3/2 e −2Rσ w 2
Ω T
∫∫
≤ C e −2Rσ f 2 + C
Ω T
∫∫
ω T
ρ 3 e −2Rσ w 2 ,
which implies the desired observability inequality, and to estimate
the observability cost with respect to the degeneracy parameter :
∫
Ω
w(0, x) 2 dx ≤ 2 T
∫ 3T /4
T /4
∫
Ω
w(t, x) 2 dx
∫∫
≤ C(Ω, ω, T , α) ρ 3/2 e −2Rσ w 2 ≤ · · ·
Ω T
25/26