Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

TL;DR

This paper employs symmetry group methods to derive explicit similarity and non-similarity solutions for a multi-dimensional semilinear heat equation, revealing analytical behaviors related to blow-up and dispersion.

Contribution

It introduces an ansatz-based symmetry analysis technique to obtain explicit solutions for a complex multi-dimensional heat equation, surpassing standard reduction methods.

Findings

Explicit similarity solutions derived.

Non-similarity solutions with blow-up and dispersion behaviors found.

Standard reduction methods are insufficient for explicit solutions.

Abstract

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.