On a family of exact solutions for a nonlinear diffusion equation

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.