A note on the symmetry group and perturbation algebra of a parabolic partial differential equation

Abstract

In this paper, the symmetry group of the parabolic partial differential equation ∂tf=Af is focused on (A is an analytic second-order elliptic linear differential operator on Rn). Following Olver [Applications of Lie Groups to Differential Equations (Springer, New York, 1986)], Ovsjannikov [Group Analysis of Differential Equations (Academic, New York, 1982)], and Rosencrans [J. Math. Anal. Appl. 56, 317 (1976); 61, 537 (1977)], the Lie algebra of infinitesimal symmetries is studied. It will be proven that it is finite dimensional (putting apart the trivial symmetries due to the linearity of the partial differential equation) by relating it to the Lie algebra of infinitesimal homothetic transformations of a Riemannian manifold (Rn,a), thus generalizing results of Ovsjannikov on elliptic equations and making use of the same Riemannian geometric techniques. Thanks to these techniques, the perturbation algebra of A, introduced by Rosencrans, is also shown to be related to geometric characteristics of (Rn,a).