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.
Versions
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BibTeX
@misc{cherniha2026multidimensional,
title = {{Multidimensional parabolic equations with a logarithmic nonlinearity admitting exceptional Lie symmetries}},
author = {Roman Cherniha and John R. King},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27810},
eprint = {2609.27810},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27810},
}
RIS
TY - UNPB
TI - Multidimensional parabolic equations with a logarithmic nonlinearity admitting exceptional Lie symmetries
AU - Roman Cherniha
AU - John R. King
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.27810
UR - https://arxiv.org/abs/2609.27810
AB - 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.
ER -