Siberian Mathematical Journal

, Volume 60, Issue 1, pp 108–113 | Cite as

Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4

  • M. V. KuznetsovEmail author


Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where \(\Delta=X_1^2+X_2^2\) is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation).


sub-Laplacian nilpotent Lie group extension method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Minakshisundaram S. and Pleijel Å., “Some properties of eigenfunctions of the Laplace operator on Riemannian manifolds,” Canad. J. Math., vol. 1, 242–256 (1949).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben Arous G., “Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut–locus,” Ann. Sci. École Norm. Sup. (4), vol. 21, No. 3, 307–331 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben Arous G., “Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale,” Ann. Inst. Fourier (Grenoble), vol. 39, No. 1, 73–99 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Craddock M. and Lennox K., “Lie group symmetries as integral transforms of fundamental solutions,” J. Differ. Equ., vol. 232, 652–674 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ovsyannikov L. V., Group Analysis of Differential Equations, Academic Press, New York (1982).Google Scholar
  6. 6.
    Boscain U., Gauthier J.–P., and Rossi F., “Hypoelliptic heat kernel over 3–step nilpotent Lie groups,” J. Math. Sci., vol. 199, No. 6, 614–628 (2014).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations