Nonlocal porous medium equation: Barenblatt profiles and other weak solutions

Piotr Biler (1) , Cyril Imbert (2) , Grzegorz Karch (1)

1 Instytut Matematyczny
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles --- the well-known solutions of the classical porous medium equation.

Mots clés: porous medium equation; nonlocal equation; hypercontractivity; self-similar solutions

Domaines: Equations aux dérivées partielles [math.AP]

Piotr Biler, Cyril Imbert, Grzegorz Karch. Nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Archive for Rational Mechanics and Analysis, 2015, 215 (2), pp.497-529. ⟨10.1007/s00205-014-0786-1⟩. ⟨hal-00795420⟩