JournalsjemsVol. 16, No. 4pp. 769–803

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type cover

Abstract

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form u(x,t) = t^{–\alpha} f(|x| t^{–\beta}) with suitable α and β. As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection principle.

Cite this article

Juan Luis Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 769–803

DOI 10.4171/JEMS/446