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Higher-order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces : the critical case

Vishvesh Kumar (UGent) , Shyam Swarup Mondal, Michael Ruzhansky (UGent) and Berikbol Torebek (UGent)
(2025) JOURNAL OF EVOLUTION EQUATIONS. 25(3).
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Abstract
Let G be a graded Lie group with homogeneous dimension . In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator R of homogeneous degree on G with power type nonlinearity and initial data taken from negative order homogeneous Sobolev space H-gamma(G), gamma >0, for the critical exponent case p=1+(2 nu)/(Q+2 gamma). We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on R-n, and more generally, on stratified Lie groups.
Keywords
Graded Lie groups, Rockland operators, Semilinear damped wave equation, Critical exponent, Negative order Sobolev spaces, Global existence, CRITICAL EXPONENT, GLOBAL EXISTENCE, HEISENBERG-GROUP, NONEXISTENCE, INEQUALITIES

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Citation

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MLA
Kumar, Vishvesh, et al. “Higher-Order Hypoelliptic Damped Wave Equations on Graded Lie Groups with Data from Negative Order Sobolev Spaces : The Critical Case.” JOURNAL OF EVOLUTION EQUATIONS, vol. 25, no. 3, 2025, doi:10.1007/s00028-025-01082-w.
APA
Kumar, V., Mondal, S. S., Ruzhansky, M., & Torebek, B. (2025). Higher-order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces : the critical case. JOURNAL OF EVOLUTION EQUATIONS, 25(3). https://doi.org/10.1007/s00028-025-01082-w
Chicago author-date
Kumar, Vishvesh, Shyam Swarup Mondal, Michael Ruzhansky, and Berikbol Torebek. 2025. “Higher-Order Hypoelliptic Damped Wave Equations on Graded Lie Groups with Data from Negative Order Sobolev Spaces : The Critical Case.” JOURNAL OF EVOLUTION EQUATIONS 25 (3). https://doi.org/10.1007/s00028-025-01082-w.
Chicago author-date (all authors)
Kumar, Vishvesh, Shyam Swarup Mondal, Michael Ruzhansky, and Berikbol Torebek. 2025. “Higher-Order Hypoelliptic Damped Wave Equations on Graded Lie Groups with Data from Negative Order Sobolev Spaces : The Critical Case.” JOURNAL OF EVOLUTION EQUATIONS 25 (3). doi:10.1007/s00028-025-01082-w.
Vancouver
1.
Kumar V, Mondal SS, Ruzhansky M, Torebek B. Higher-order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces : the critical case. JOURNAL OF EVOLUTION EQUATIONS. 2025;25(3).
IEEE
[1]
V. Kumar, S. S. Mondal, M. Ruzhansky, and B. Torebek, “Higher-order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces : the critical case,” JOURNAL OF EVOLUTION EQUATIONS, vol. 25, no. 3, 2025.
@article{01JX3J4165H3RFTGT4V4RZB3HF,
  abstract     = {{Let G be a graded Lie group with homogeneous dimension . In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator R of homogeneous degree on G with power type nonlinearity and initial data taken from negative order homogeneous Sobolev space H-gamma(G), gamma >0, for the critical exponent case p=1+(2 nu)/(Q+2 gamma). We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on R-n, and more generally, on stratified Lie groups.}},
  articleno    = {{58}},
  author       = {{Kumar, Vishvesh and Mondal, Shyam Swarup and Ruzhansky, Michael and Torebek, Berikbol}},
  issn         = {{1424-3199}},
  journal      = {{JOURNAL OF EVOLUTION EQUATIONS}},
  keywords     = {{Graded Lie groups,Rockland operators,Semilinear damped wave equation,Critical exponent,Negative order Sobolev spaces,Global existence,CRITICAL EXPONENT,GLOBAL EXISTENCE,HEISENBERG-GROUP,NONEXISTENCE,INEQUALITIES}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{29}},
  title        = {{Higher-order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces : the critical case}},
  url          = {{http://doi.org/10.1007/s00028-025-01082-w}},
  volume       = {{25}},
  year         = {{2025}},
}
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