Some properties of the equation of fast diffusion and its multidimensional exact solutions

TL;DR

This paper proves invariance properties of the fast diffusion equation in 2D, constructs new multidimensional exact solutions dependent on harmonic functions, and applies these results to related quasilinear and elliptic systems.

Contribution

It introduces new invariance results and exact solutions for the fast diffusion equation and related systems, expanding analytical tools in this area.

Findings

Proved invariance of the 2D fast diffusion equation.

Constructed new multidimensional solutions based on harmonic functions.

Extended results to systems in semiconductor theory.

Abstract

The invariance for the equation of fast diffusion in the 2D coordinate space has been proved, and its reduction to the 1D (with respect to the spatial variable) analog is demonstrated. On the basis of these results, new exact multi-dimensional solutions, which are dependent on arbitrary harmonic functions, are constructed. As a result, new exact solutions of the well-known Liouville equation - the steady-state analog for the fast diffusion equation with the linear source - have been obtained. Some generalizations for the systems of quasilinear parabolic equations, as well as systems of elliptic equations with Poisson interaction, which are applied in the theory of semiconductors, are considered.