Solving a nonlinear variation of the heat equation: self-similar solutions of the second kind and other results

Abstract

This paper studies regular self-similar solutions of the following diffusion equation $$\begin{aligned} u_{t}+\gamma \vert u_{t} \vert =\Delta u\quad \text {in}\ \mathbb {R}^{N}\times ]0,\infty [, \end{aligned}$$ where $$-1<\gamma <1$$ . The analysis is focused on radial symmetric solutions $$u(x,t)=t^{-\alpha /2}f(\eta )$$ with $$\alpha >0$$ and $$\eta =\Vert x\Vert /\sqrt{t}$$ . Closed representation is obtained in terms of confluent hypergeometric functions. Employing specific properties of these special functions, oscillatory and symptotic aspects of f are obtained. It is demonstrated that such features are governed by increasing and unbounded sequences of exponents $$\alpha _{0}<\alpha _{1}<\cdots $$ , as in other diffusion equations. These exponents are determined by solving a system of transcendental equations related to specific roots of Kummer and Tricomi functions. As these cannot be determined using dimensional analysis, it is concluded that they are anomalous. For each exponent $$\alpha _{k}$$ , linear approximation when $$\gamma $$ is close to zero is also presented. Finally, relationships with previous results as well as an extension to other fully nonlinear parabolic equations are discussed.