Observability and stabilizability for Grushin-Type parabolic equations in the whole space
Abstract
We study observability and stabilization for the whole-space Grushin heat equation ∂t u−Δγu=0, Δγ:=Δx+|x|2γΔy, (t,x,y)∈(0,T)×ℝn×ℝm, γ>0. For product sets E=E1× E2, we identify three observability regimes in which thickness, distance from the degeneracy set {x=0}, and local density near that set play different roles. (i) In the subcritical regime 0<γ<1, the observability inequality holds for every T>0 if and only if both E1 and E2 are thick; hence Tmin(E)=0. (ii) At the critical exponent γ=1, if E2=ℝm and E1⊂ℝn is thick, we determine the exact minimal observable time: Tmin(E)=aess(E1)2/2n, aess(E1):=*ess infx∈ E1|x|. Thus, the threshold depends on the essential distance of E1 from the degeneracy set. (iii) In the supercritical regime γ>1, we consider n=1, E2=ℝm and thick sets E1⊂ℝ. If aess(E1)>0, observability fails for every T>0, so that Tmin(E)=+∞. If aess(E1)=0, observability depends on the density of E1 near $0$, measured by DE1(r):=log (2r/(|E1∩(−r,r)|)), LE1:=limsupr↓0r2DE1(r). If LE1=0, then Tmin(E)=0; if 0<LE1<∞, a finite sufficient observation time exists. Moreover, observability at time T requires limsupr↓0r2(1+γ)/(γ−1)DE1(r)≤ CT,γ. In contrast, for every γ>0, exponential and complete stabilizability hold if and only if both E1 and E2 are thick. The stabilization criterion is therefore independent of γ.
Related
- Observability properties of the singular Grushin equation
- Sharp Observability Costs for Strongly Coupled Parabolic Systems with Vanishing Principal Coupling
- Observability from measurable sets for strongly coupled parabolic systems
- Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes
- Observability for wave equations with critically singular potentials
BibTeX
@misc{huang2026observability,
title = {{Observability and stabilizability for Grushin-Type parabolic equations in the whole space}},
author = {Shanlin Huang and Gengsheng Wang and Jiaqi Yu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05821},
eprint = {2611.05821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05821},
}
RIS
TY - UNPB
TI - Observability and stabilizability for Grushin-Type parabolic equations in the whole space
AU - Shanlin Huang
AU - Gengsheng Wang
AU - Jiaqi Yu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05821
UR - https://arxiv.org/abs/2611.05821
AB - We study observability and stabilization for the whole-space Grushin heat equation ∂_t u−Δ_γu=0, Δ_γ:=Δ_x+|x|^2γΔ_y, (t,x,y)∈(0,T)×ℝ^n×ℝ^m, γ>0. For product sets E=E_1× E_2, we identify three observability regimes in which thickness, distance from the degeneracy set {x=0}, and local density near that set play different roles. (i) In the subcritical regime 0<γ<1, the observability inequality holds for every T>0 if and only if both E_1 and E_2 are thick; hence T_min(E)=0. (ii) At the critical exponent γ=1, if E_2=ℝ^m and E_1⊂ℝ^n is thick, we determine the exact minimal observable time: T_min(E)=a_ess(E_1)^2/2n, a_ess(E_1):=*ess inf_x∈ E_1|x|. Thus, the threshold depends on the essential distance of E_1 from the degeneracy set. (iii) In the supercritical regime γ>1, we consider n=1, E_2=ℝ^m and thick sets E_1⊂ℝ. If a_ess(E_1)>0, observability fails for every T>0, so that T_min(E)=+∞. If a_ess(E_1)=0, observability depends on the density of E_1 near $0$, measured by D_E_1(r):=log (2r/(|E_1∩(−r,r)|)), L_E_1:=limsup_r↓0r^2D_E_1(r). If L_E_1=0, then T_min(E)=0; if 0<L_E_1<∞, a finite sufficient observation time exists. Moreover, observability at time T requires limsup_r↓0r^2(1+γ)/(γ−1)D_E_1(r)≤ C_T,γ. In contrast, for every γ>0, exponential and complete stabilizability hold if and only if both E_1 and E_2 are thick. The stabilization criterion is therefore independent of γ.
ER -