Multidimensional parabolic equations with a logarithmic nonlinearity admitting exceptional Lie symmetries

Abstract

Nonlinear reaction-diffusion equations arise in a very wide range of applications. Here we address a specific multidimensional class in which the source term is non-autonomous in time, this choice being motivated both by its relevance to a broad class of (autonomous) `near-linear' reaction-diffusion equations and by the unusually rich nature of the Lie symmetries that results (including for arbitrary time-dependence, and with four even more exceptional cases arising). The class of non-autonomous equation analysed is derived by an asymptotic approach, with the Lie symmetry classification and a further analysis subsequently carried out. Both Lie and non-Lie reductions to ordinary differential equations are also noted and relevant exact solutions are constructed.

arXiv comment19 pages