Eventual self-similarity of solutions for the diffusion equation with nonlinear absorption and a point source

TL;DR

This paper investigates the long-term behavior of solutions to reaction-diffusion equations with nonlinear absorption and a point source, demonstrating self-similarity in various dimensions and nonlinearities.

Contribution

It extends the understanding of self-similar solution behavior from one dimension to higher dimensions under specific nonlinear conditions.

Findings

Solutions in 2D approach a self-similar form over time.

In 3D and higher, self-similarity holds for nonlinear powers up to the Serrin critical exponent.

Existence of ultra-singular solutions is established through a variational approach.

Abstract

This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point source. In one space dimension, solutions of these problems with zero initial data are known to approach the stationary solution in an asymptotically self-similar manner. Here we show that this conclusion remains true in two space dimensions, while in three and higher dimensions the same conclusion holds true for all powers of the nonlinearity not exceeding the Serrin critical exponent. The analysis requires dealing with solutions that contain a persistent singularity and involves a variational proof of existence of ultra-singular solutions, a special class of self-similar solutions in the considered problem.