Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order
- Author
- Aparajita Dasgupta, Vishvesh Kumar (UGent) , Shyam Swarup Mondal and Michael Ruzhansky (UGent)
- Organization
- Project
- Abstract
- In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian L on the Heisenberg group H^n with power type nonlinearity |u|^p and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space H˙L^-γ(H^n), γ > 0, on H^n. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent pcrit(Q,γ) = 1+ 4/(Q+2γ) , for γ ∈ (0, Q/2), where Q := 2n+2 is the homogeneous dimension of H^n. More precisely, we establish - A global-in-time existence of small data Sobolev solutions of lower regularity for in the energy evolution space; - A finite time blow-up of weak solutions for under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.
- Keywords
- Heisenberg group, Semilinear damped wave equation, Critical exponent, Negative order Sobolev spaces, Global existence, Finite blow-up, LIFE-SPAN, CRITICAL EXPONENT, GLOBAL-SOLUTIONS, EXISTENCE, NONEXISTENCE, INEQUALITIES, BLOW
Downloads
-
s00028-024-00976-5-6.pdf
- full text (Published version)
- |
- open access
- |
- |
- 694.06 KB
Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-01JPAADAE73AJT6ECY1AZV5Q71
- MLA
- Dasgupta, Aparajita, et al. “Semilinear Damped Wave Equations on the Heisenberg Group with Initial Data from Sobolev Spaces of Negative Order.” JOURNAL OF EVOLUTION EQUATIONS, vol. 24, no. 3, 2024, doi:10.1007/s00028-024-00976-5.
- APA
- Dasgupta, A., Kumar, V., Mondal, S. S., & Ruzhansky, M. (2024). Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order. JOURNAL OF EVOLUTION EQUATIONS, 24(3). https://doi.org/10.1007/s00028-024-00976-5
- Chicago author-date
- Dasgupta, Aparajita, Vishvesh Kumar, Shyam Swarup Mondal, and Michael Ruzhansky. 2024. “Semilinear Damped Wave Equations on the Heisenberg Group with Initial Data from Sobolev Spaces of Negative Order.” JOURNAL OF EVOLUTION EQUATIONS 24 (3). https://doi.org/10.1007/s00028-024-00976-5.
- Chicago author-date (all authors)
- Dasgupta, Aparajita, Vishvesh Kumar, Shyam Swarup Mondal, and Michael Ruzhansky. 2024. “Semilinear Damped Wave Equations on the Heisenberg Group with Initial Data from Sobolev Spaces of Negative Order.” JOURNAL OF EVOLUTION EQUATIONS 24 (3). doi:10.1007/s00028-024-00976-5.
- Vancouver
- 1.Dasgupta A, Kumar V, Mondal SS, Ruzhansky M. Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order. JOURNAL OF EVOLUTION EQUATIONS. 2024;24(3).
- IEEE
- [1]A. Dasgupta, V. Kumar, S. S. Mondal, and M. Ruzhansky, “Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order,” JOURNAL OF EVOLUTION EQUATIONS, vol. 24, no. 3, 2024.
@article{01JPAADAE73AJT6ECY1AZV5Q71,
abstract = {{In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian L on the Heisenberg group H^n with power type nonlinearity |u|^p and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space H˙L^-γ(H^n), γ > 0, on H^n. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent pcrit(Q,γ) = 1+ 4/(Q+2γ) , for γ ∈ (0, Q/2), where Q := 2n+2 is the homogeneous dimension of H^n. More precisely, we establish
- A global-in-time existence of small data Sobolev solutions of lower regularity for
in the energy evolution space;
- A finite time blow-up of weak solutions for
under certain conditions on the initial data by using the test function method.
Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.}},
articleno = {{51}},
author = {{Dasgupta, Aparajita and Kumar, Vishvesh and Mondal, Shyam Swarup and Ruzhansky, Michael}},
issn = {{1424-3199}},
journal = {{JOURNAL OF EVOLUTION EQUATIONS}},
keywords = {{Heisenberg group,Semilinear damped wave equation,Critical exponent,Negative order Sobolev spaces,Global existence,Finite blow-up,LIFE-SPAN,CRITICAL EXPONENT,GLOBAL-SOLUTIONS,EXISTENCE,NONEXISTENCE,INEQUALITIES,BLOW}},
language = {{eng}},
number = {{3}},
pages = {{35}},
title = {{Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order}},
url = {{http://doi.org/10.1007/s00028-024-00976-5}},
volume = {{24}},
year = {{2024}},
}
- Altmetric
- View in Altmetric
- Web of Science
- Times cited: