Global actions of Lie symmetries for the nonlinear heat equation (Q837113)

Standard Lie symmetry theory of differential equations is mainly concerned with local Lie groups (or often even only with infinitesimal symmetries). Following an approach developed by \textit{M. Craddock} [J. Differ. Equations 116, No. 1, 202--247 (1995; Zbl 0845.35020), ibid. 166, No. 1, 107--131 (2000; Zbl 0962.35010)], the author shows how one can obtain for the nonlinear heat equation \(u_t=(k(u)u_x)_x\) a global action of the symmetry group by restricting to a class of smooth functions naturally arising via representation theory. Modulo equivalence transformations, one must consider four different cases of the function \(k(u)\). It turns out that in three of them a linear action arises; only for \(k(u)=e^u\) a nonlinear action is obtained.