$\mathbb{R}^n$高阶线性抛物方程的唯一延拓性与能观性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (6): 2196-2206.

$\mathbb{R}^n$高阶线性抛物方程的唯一延拓性与能观性

郑国杰¹, 叶笼汉¹, 朱理², 于欣³^,^*()

¹ 宁波财经学院机械工程学院浙江宁波 315175
² 武汉工程大学数理学院武汉 430205
³ 浙大宁波理工学院信息处理与智能计算研究所浙江宁波 315100

收稿日期:2025-07-28 修回日期:2026-01-23 出版日期:2026-12-26 发布日期:2026-08-14
通讯作者: 于欣 E-mail:yuxin@zju.edu.cn
基金资助:
国家自然科学基金(12171431);浙江省自然科学基金(LY24A010014);浙江省自然科学基金(LZ21A010001)

Unique Continuation and Observability of Higher-Order Linear Parabolic Equation in $\mathbb{R}^n$

Guojie Zheng¹, Longhan Ye¹, Li Zhu², Xin Yu³^,^*()

¹ College of Mechanical Engineering, Ningbo University of Finance & Economics, Zhejiang Ningbo 315175
² School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan 430205
³ Institute of Information Processing and Intelligent Computing, Ningbo Tech University, Zhejiang Ningbo 315100

Received:2025-07-28 Revised:2026-01-23 Online:2026-12-26 Published:2026-08-14
Contact: Xin Yu E-mail:yuxin@zju.edu.cn
Supported by:
NSFC(12171431);Zhejiang Provincial NSF(LY24A010014);Zhejiang Provincial NSF(LZ21A010001)

摘要/Abstract

摘要：

该研究旨在探讨全空间中高阶线性抛物方程的唯一延拓性和能观性问题. 在此研究过程中, 首先讨论厚集与全空间上谱不等式之间的关系; 其次, 运用谱不等式研究该类方程的解所满足的一类 Hölder 型插值不等式; 借助于 Hölder 型插值不等式可证明全空间高阶抛物方程的定量唯一延拓性质; 最后, 利用 Lebeau-Robbiano 方法证明全空间中高阶线性抛物方程的能观性估计.

关键词: 能观性估计, 高阶线性抛物方程, 唯一延拓性, 谱不等式

Abstract:

This paper study the observability and unique continuation property of higher-order linear parabolic equations in the whole space. First, the relationship between thick set and spectral inequality is discussed. Secondly, by using spectral inequality, a class of Hölder-type interpolation inequality for the solution of this equation is investigated. Furthermore, with the help of this inequality, the quantitative unique continuation property of this equation in the whole space can be obtained. Finally, the observability estimate for linear higher-order parabolic equations in $\mathbb{R}^n$ is proved by Lebeau-Robbiano strategy.

Key words: observability estimate, higher-order linear parabolic equation, unique continuation property, spectral inequality

中图分类号:

O231.4

郑国杰, 叶笼汉, 朱理, 于欣. $\mathbb{R}^n$高阶线性抛物方程的唯一延拓性与能观性[J]. 数学物理学报, 2026, 46(6): 2196-2206.

Guojie Zheng, Longhan Ye, Li Zhu, Xin Yu. Unique Continuation and Observability of Higher-Order Linear Parabolic Equation in $\mathbb{R}^n$[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2196-2206.

参考文献 19

[1]	Barbatis G. Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients. J Differ Equat, 2001, 174(2): 442-463 doi: 10.1006/jdeq.2000.3940
[2]	Cholewa J W, Rodriguez-Bernal A. Linear and semilinear higher order parabolic equations in $\mathbb{R}^N$. Nonlinear Anal-Theor, 2012, 75(1): 194-210 doi: 10.1016/j.na.2011.08.022
[3]	Ding Y, Sun X C. Strichartz estimates for parabolic equations with higher order differential operators. Sci China Math, 2015, 58(5): 1047-1062 doi: 10.1007/s11425-014-4869-0
[4]	Bernis F, Friedman A. Higher order nonlinear degenerate parabolic equations. J Differ Equat, 1990, 83(1): 179-206 doi: 10.1016/0022-0396(90)90074-Y
[5]	Bertozzi A L, Pugh M. The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions. Comm Pure Appl Math, 1996, 49(2): 85-123 doi: 10.1002/(ISSN)1097-0312
[6]	Lysaker M, Lundervold A, Tai X C. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE T Image Process, 2003, 12(12): 1579-1590 doi: 10.1109/TIP.2003.819229 pmid: 18244712
[7]	Laurençot P, Walker C. The time singular limit for a fourth-order damped wave equation for MEMS//Escher J, Schrohe E, Seiler J, Walker C. Elliptic and Parabolic Equations. Cham: Springer, 2015, 233-246
[8]	Coron J. Control and Nonlinearity. Providence: AMS, 2007
[9]	Camliyurt G, Kukavica I. Quantitative unique continuation for a parabolic equation. Indiana Univ Math J, 2018, 67(2): 657-678 doi: 10.1512/iumj.2018.67.7283
[10]	Lin F. A uniqueness theorem for parabolic equations. Comm Pure Appl Math, 1990, 43(1): 127-136 doi: 10.1002/cpa.v43:1
[11]	Phung K D, Wang G. An observability estimate for the parabolic equations from a measurable set in time and its applications. J Eur Math Soc, 2013, 15: 681-703 doi: 10.4171/jems
[12]	Zheng G, Wang T. Unique continuation and observability for a semi-linear parabolic system. ESAIM Control Optim Calc Var, 2024, 30: Art 27
[13]	Zhou Z. Observability estimate and null controllability for one-dimensional fourth order parabolic equation. Taiwan J Math, 2012, 16(6): 1991-2017
[14]	Guerrero S, Kassab K. Carleman estimate and null controllability of a fourth order parabolic equation in dimension $N\geq2$. J Math Pures Appl, 2019, 121: 135-161 doi: 10.1016/j.matpur.2018.04.004
[15]	Lü Q, Wang Y. Null controllability for stochastic fourth order parabolic equations. SIAM J Control Optim, 2022, 60(3): 1563-1590 doi: 10.1137/22M1472620
[16]	Li F, You B. Global Carleman estimates for a fourth order parabolic equation and application to null controllability. Math Control Relat Fields, 2026, 17: 120-143 doi: 10.3934/mcrf.2025031
[17]	Taylor M E. Partial Differential Equations II. NewYork: Springer, 1997
[18]	Wang G, Wang M, Zhang C, Zhang Y. Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$. J Math Pures Appl, 2019, 126: 144-194 doi: 10.1016/j.matpur.2019.04.009
[19]	Egidi M, Veselić I. Sharp geometric condition for null-controllability of the heat equation on $\mathbb{R}^d$ and consistent estimates on the control cost. Arch Math, 2018, 111: 85-99 doi: 10.1007/s00013-018-1185-x