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

Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence, France

^* Corresponding author: yunlei.wang@math.u-bordeaux.fr

Received: 12 March 2024
Accepted: 29 July 2024

Abstract

We study the relation between propagation of smallness in the plane and control for heat equations. The former has been proved by Zhu Preprint arXiv:2304.09800 (2023) 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 ∈ L_loc^∞(ℝ). As a key ingredient, new spectral inequalities are established. The control set Ω that we consider satisfy

| Ω∩[ x-L〈 x 〉^-s,x+L〈 x 〉^-s ] |≥γ^{〈 x 〉τ}2L〈 x 〉^-s

for some γ ∈ (0, 1), L > 0, τ, s ≥ 0, and ⟨x⟩ := (1 + |x|²)^1/2. In particular, the null-controllability result for the case of thick sets that allow the decay of the density (i.e., s = 0 and τ ≥ 0) is included. These extend the results in [J. Zhu and J. Zhuge Preprint arXiv:2301.12338 (2023)] from Ω being the union of equidistributive open sets to thick sets in the 1-dimensional case, and in [P. Su et al. Preprint arXiv:2309.00963 (2023)] from bounded potentials to certain unbounded ones.