On a family of exact solutions for a nonlinear diffusion equation

TL;DR

This paper thoroughly analyzes similarity solutions for a nonlinear diffusion equation using special functions, revealing new asymptotic behaviors and establishing a connection between solution profiles and zeros of hypergeometric functions.

Contribution

It provides explicit representations of solutions in terms of hypergeometric functions and introduces new methods to analyze their asymptotic and oscillatory behaviors.

Findings

Existence of an unbounded sequence of similarity exponents with Gaussian decay.

Explicit solutions expressed via Kummer and Tricomi functions.

Connection between zeros of special functions and solution profiles.

Abstract

We present a complete description of the similarity solutions $u_{\alpha}(x,t)=t^{-\alpha/2}f(\Vert x \Vert/\sqrt{t};\alpha)$ for the following nonlinear diffusion equation $$ u_{t}+\gamma\vert u_{t} \vert =\Delta u\qquad(-1<\gamma<1) $$ The behaviors of these solutions are obtained through the explicit representation of $f(\eta;\alpha)$, in terms of Kummer and Tricomi functions. Considering results about confluent hypergeometric functions, new methods to describe asymptotic and oscillatory behaviors of the similarity solutions are obtained. We prove that there exists an increasing and unbounded sequence of positive similarity exponents such that the associated profile $f$ has a gaussian rate decay. These special similarity exponents are related with the zeros of Kummer and Tricomi functions. Finally, we indicate how to extend our results on more general nonlinear diffusion equations.