Null-controllability for weakly dissipative heat-like equations

We study the null-controllability properties of heat-like equations posed on the whole Euclidean space ${\mathbb{R}}^n$. These evolution equations are associated with Fourier multipliers of the form $\rho (\mathrm{\mid }{D}_x\mathrm{\mid })$, where $\rho \text{\hspace{0.17em}⁣}:[0,+\mathrm{\infty })\to \mathbb{C}$ is a measurable function such that $\mathrm{\Re }\rho$ is bounded from below. We consider the ``weakly dissipative’’ case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho (\xi )={\xi }^s$ in the regime $s\in (0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho (\xi )=o(\xi )$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$.Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.