Radial equivalence and applications to the qualitative theory for a class of non-homogeneous reaction-diffusion equations

TL;DR

This paper introduces transformations for radially symmetric solutions of non-homogeneous reaction-diffusion equations, enabling new qualitative insights and the construction of self-similar solutions, thereby advancing the theoretical understanding of these equations.

Contribution

It develops novel transformations for radially symmetric solutions, enhancing the analysis and construction of self-similar solutions in non-homogeneous reaction-diffusion equations.

Findings

Derived general qualitative properties of solutions.

Constructed self-similar solutions as dynamic patterns.

Extended mappings to the semilinear case m=1.

Abstract

Some transformations acting on radially symmetric solutions to the following class of non-homogeneous reaction-diffusion equations $$ |x|^{\sigma_1}\partial_tu=\Delta u^m+|x|^{\sigma_2}u^p, \qquad (x,t)\in\real^N\times(0,\infty), $$ which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here $m\geq1$, $p\geq1$, $N\geq1$ and $\sigma_1$, $\sigma_2$ real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self-similar solutions which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case $m=1$.