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

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, X_j] = X_j+1 (where X_n+1 = 0) and [X_j, X_k] = 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).