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

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Abstract

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).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    M. V. Kuznetsov

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  1. M. V. Kuznetsov
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Correspondence to M. V. Kuznetsov.

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

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Translated from Sibirskii Matematicheskii Zhurnal, vol. 60, no. 1, pp. 141–148, January–February, 2019; DOI: 10.17377/smzh.2019.60.112.

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Kuznetsov, M.V. Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4. Sib Math J 60, 108–113 (2019). https://doi.org/10.1134/S0037446619010129

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  • DOI: https://doi.org/10.1134/S0037446619010129

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