The null controllability for a singular heat equation with variable coefficients

Impact Factor:1.1

DOI number:10.1080/00036811.2020.1769076

Journal:Applicable Analysis

Key Words:Singular heat equation; null controllability; Carleman estimate; observability

Abstract:The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: ∂_tu(x, t) − div(p(x)∇u (x, t)) − (μ/|x|^2) u (x, t) = f (x, t). Here μ is a real constant. It was proved in the paper of Goldstein and Zhang (8) that the equation is well-posedness when 0 ≤ μ ≤ p_1(n − 2)^2/4, and in this paper, we mainly consider the case 0 ≤ μ<(p^2_1/p_2)(n −2)^2/4, where p_1, p_2 are two positive constants which satisfy: 0 < p_1 ≤ p(x) ≤ p_2, ∀ x ∈ Ω. We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996–2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317] to the system. We obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case μ > p_2(n − 2)^2/4. We consider a sequence of regularized potentials μ/(|x|^2 + ε^2), and prove that we cannot stabilize the corresponding systems uniformly with respect to ε> 0.

Co-author:Shumin Li

First Author:Xue Qin

Indexed by:Journal paper

Document Code:000537939600001

Discipline:Natural Science

Document Type:J

Volume:101

Issue:3

Page Number:1052-1076

ISSN No.:0003-6811

Translation or Not:no

Date of Publication:2022-02-11

Included Journals:SCI

Links to published journals:https://www.tandfonline.com/eprint/CJUBMW2MGIDP5EIXNUIC/full?target=10.1080/00036811.2020.1769076