Quantitative 2D propagation of smallness and control for 1D heat equations with power growth potentials

We study the relation between propagation of smallness in the plane and control for heat equations. The former has been proved by Zhu \cite{zhu2023remarks} who showed how the value of solutions in some small set propagates to a larger domain. By reviewing his proof, we establish a quantitative version with the explicit dependence of parameters. Using this explicit version, we establish new exact null-controllability results of 1D heat equations with any nonnegative power growth potentials $V\in L^\infty_{\mathrm{loc}}(\mathbb{R})$. As a key ingredient, new spectral inequalities are established. The control set $\Omega$ that we consider satisfy \begin{equation*} \left|\Omega\cap [x-L\langle x\rangle ^{-s},x+L\langle x\rangle ^{-s}]\right|\ge \gamma^{\langle x\rangle^\tau}2L\langle x\rangle^{-s} \end{equation*} for some $\gamma\in(0,1)$, $L>0$, $\tau,s\ge 0$, and $\langle x\rangle:=(1+|x|^2)^{1 /2} $. In particular, the null-controllability result for the case of thick sets that allow the decay of the density (\textit{i.e.}, $s=0$ and $\tau\ge 0$) is included. These extend the results in \cite{zhu2023spectral} from $\Omega$ being the union of equidistributive open sets to thick sets in the 1-dimensional case, and in \cite{su2023quantitative} from bounded potentials to certain unbounded ones.