Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains

We study the null controllability of the linear parabolic equation involving the Grushin operator Gs=Δx+|x|2sΔy(s>0), in the domain Ω=(−1,1)N1×(0,1)N2⊂RN1×RN2,N1,N2≥1Ω=(−1,1)N1×(0,1)N2⊂RN1×RN2,N1,N2≥1, under an additive control supported in an open subset ωω of ΩΩ. We prove that the equation is null controllable in any positive time for s<1s<1 and not null controllable for s>1s>1. When s=1s=1, a positive minimal time is required for null controllability.