Decay estimates for the structural damped wave equations on Heisenberg groups

The Anh Bui; The Quan Bui; Xuan Thinh Duong

doi:10.1007/s00030-025-01073-3

Decay estimates for the structural damped wave equations on Heisenberg groups

The Anh Bui^*, The Quan Bui, Xuan Thinh Duong

^*Corresponding author for this work

School of Mathematical and Physical Sciences

Research output: Contribution to journal › Article › peer-review

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Abstract

In this paper, we study the structural damped wave equation on the Heisenberg group Hn and its solution u(t,η), t > 0,η∈Hn. We prove decay estimates including L1-L2 estimates and L1-L∞ estimates for the solution u(t,η) and its higher-order time derivative and horizontal gradient ∂_t^ℓ∇_Hn^ku(t,η). Our approach relies on some detailed estimates using the group Fourier transform and the functional calculus of the sub-Laplacian. The obtained results are not only new but also improve previously known results.

Original language	English
Article number	84
Pages (from-to)	1-34
Number of pages	34
Journal	Nonlinear Differential Equations and Applications
Volume	32
Issue number	5
DOIs	https://doi.org/10.1007/s00030-025-01073-3
Publication status	Published - Sept 2025

Bibliographical note

Keywords

Decay estimate
Functional calculus
Group Fourier transform
Heisenberg group
Structural damped wave equation

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10.1007/s00030-025-01073-3Licence: CC BY

Publisher version (open access)Final published version, 562 KBLicence: CC BY

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title = "Decay estimates for the structural damped wave equations on Heisenberg groups",

abstract = "In this paper, we study the structural damped wave equation on the Heisenberg group Hn and its solution u(t,η), t > 0,η∈Hn. We prove decay estimates including L1-L2 estimates and L1-L∞ estimates for the solution u(t,η) and its higher-order time derivative and horizontal gradient ∂tℓ∇Hnk u(t,η). Our approach relies on some detailed estimates using the group Fourier transform and the functional calculus of the sub-Laplacian. The obtained results are not only new but also improve previously known results.",

keywords = "Decay estimate, Functional calculus, Group Fourier transform, Heisenberg group, Structural damped wave equation",

author = "Bui, \{The Anh\} and Bui, \{The Quan\} and Duong, \{Xuan Thinh\}",

note = "{\textcopyright} The Author(s) 2025. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.",

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doi = "10.1007/s00030-025-01073-3",

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T1 - Decay estimates for the structural damped wave equations on Heisenberg groups

AU - Bui, The Anh

AU - Bui, The Quan

AU - Duong, Xuan Thinh

N1 - © The Author(s) 2025. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

PY - 2025/9

Y1 - 2025/9

N2 - In this paper, we study the structural damped wave equation on the Heisenberg group Hn and its solution u(t,η), t > 0,η∈Hn. We prove decay estimates including L1-L2 estimates and L1-L∞ estimates for the solution u(t,η) and its higher-order time derivative and horizontal gradient ∂tℓ∇Hnk u(t,η). Our approach relies on some detailed estimates using the group Fourier transform and the functional calculus of the sub-Laplacian. The obtained results are not only new but also improve previously known results.

AB - In this paper, we study the structural damped wave equation on the Heisenberg group Hn and its solution u(t,η), t > 0,η∈Hn. We prove decay estimates including L1-L2 estimates and L1-L∞ estimates for the solution u(t,η) and its higher-order time derivative and horizontal gradient ∂tℓ∇Hnk u(t,η). Our approach relies on some detailed estimates using the group Fourier transform and the functional calculus of the sub-Laplacian. The obtained results are not only new but also improve previously known results.

KW - Decay estimate

KW - Functional calculus

KW - Group Fourier transform

KW - Heisenberg group

KW - Structural damped wave equation

UR - https://www.scopus.com/pages/publications/105008525308

UR - http://purl.org/au-research/grants/arc/DP220100285

U2 - 10.1007/s00030-025-01073-3

DO - 10.1007/s00030-025-01073-3

M3 - Article

AN - SCOPUS:105008525308

SN - 1021-9722

VL - 32

SP - 1

EP - 34

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

IS - 5

M1 - 84

ER -

Decay estimates for the structural damped wave equations on Heisenberg groups

Abstract

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DP22: Harmonic analysis of Laplacians in curved spaces

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Decay estimates for the structural damped wave equations on Heisenberg groups

Abstract

Bibliographical note

Keywords

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DP22: Harmonic analysis of Laplacians in curved spaces

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