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Sibirsk. Mat. Zh., 2019, Volume 60, Number 1, Pages 141–147 (Mi smj3065)  

Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least $4$

M. V. Kuznetsov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: Using the extension method, we study the one-parameter symmetry groups of the heat equation $\partial_{t} p=\Delta p$, where $\Delta=X_{1}^{2}+X_{2}^{2}$ is the sub-Laplacian constructed by a Goursat distribution $\operatorname{span} (\lbrace X_{1},X_{2} \rbrace)$ in $\mathbb{R}^n$, where the vector fields $X_1$ and $X_2$ satisfy the commutation relations $[X_{1},X_{j}]=X_{j+1}$ (where $X_{n+1}=0$) and $[X_{j},X_{k}]=0$ for $j \geq 1$ and $k \geq 1$. We show that there are no such groups for $n \geq 4$ (with exception of the linear transformations of solutions which are admitted by every linear equation).

Keywords: sub-Laplacian, nilpotent Lie group, extension method.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.12875.2018/12.1
The author was supported by the Ministry of Education and Science of the Russian Federation (Grant 1.12875.2018/12.1).


DOI: https://doi.org/10.33048/smzh.2019.60.112

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English version:
Siberian Mathematical Journal, 2019, 60:1, 108–113

Bibliographic databases:

UDC: 512.813.52+514.763.85
MSC: 35R30
Received: 09.04.2018
Revised: 18.07.2018
Accepted:17.10.2018

Citation: M. V. Kuznetsov, “Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least $4$”, Sibirsk. Mat. Zh., 60:1 (2019), 141–147; Siberian Math. J., 60:1 (2019), 108–113

Citation in format AMSBIB
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\paper Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least~$4$
\jour Sibirsk. Mat. Zh.
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\vol 60
\issue 1
\pages 141--147
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\crossref{https://doi.org/10.33048/smzh.2019.60.112}
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\jour Siberian Math. J.
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\pages 108--113
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