We study a porous medium equation with fractional potential pressure: ∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, for m > 1, 0 < s < 1 and u(x, t) ≥ 0. To be specific, the problem is posed for x ∈ R N , N ≥ 1, and t > 0. The initial data u(x, 0) is assumed to be a bounded function with compact support or fast decay at infinity. We establish existence of a class of weak solutions for which we determine whether, depending on the parameter m, the property of compact support is conserved in time or not, starting from the result of finite propagation known for m = 2. We find that when m ∈ [1, 2) the problem has infinite speed of propagation, while for m ∈ [2, ∞) it has finite speed of propagation. Comparison is made with other nonlinear diffusion models where the results are widely different. Résumé Vitesse de propagation finie et infinie pour deséquations du milieu poreux avec une pression fractionnaire. Nousétudions uneéquation du milieu poreux avec une pression potentielle fractionnaire: ∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, pour m > 1, 0 < s < 1 et u(x, t) ≥ 0. Le problème se pose pour x ∈ R N , N ≥ 1 et t > 0. La donnée initiale est supposée bornée avec support compact ou décroissance rapideà l'infini. Lorsque le paramètre m est variable, on obtient deux comportements différents comme suit: si m ∈ [1, 2) le problème a une vitesse de propagation infinie, alors que pour m ∈ [2, ∞), elle a une vitesse de propagation finie. On compare le résultat avec le comportement d'autres modèles de diffusion nonlinéaire qui est très différent.

We study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractional diffusion:For all 0 < s < 1 and m > m c = (N − 2s) + /N , we consider the solution of the initial-value problem with initial data having fast decay at infinity and prove that its level sets propagate exponentially fast in time, in contradiction to the traveling wave behaviour of the standard KPP case, which corresponds to putting s = 1, m = 1 and f (u) = u(1 − u). The proof of this fact uses as an essential ingredient the recently established decay properties of the self-similar solutions of the purely diffusive equation, u t + (−∆) s u m = 0. 2000 Mathematics Subject Classification. 35K57, 26A33, 35K65, 76S05, 35C06, 35C07.

We study a porous medium equation with fractional potential pressure:∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, for m > 1, 0 < s < 1 and u(x, t) ≥ 0. The problem is posed for x ∈ R N , N ≥ 1, and t > 0. The initial data u(x, 0) is assumed to be a bounded function with compact support or fast decay at infinity. We establish existence of a class of weak solutions for which we determine whether the property of compact support is conserved in time depending on the parameter m, starting from the result of finite propagation known for m = 2. We find that when m ∈ [1, 2) the problem has infinite speed of propagation, while for m ∈ [2, 3) it has finite speed of propagation. In other words m = 2 is critical exponent regarding propagation. The main results have been announced in the note [29].Other related models. Equation (CV) with s = 1/2 in dimension N = 1 has been proposed by Head [20] to describe the dynamics of dislocation in crystals. The model is written in the integrated form asThe dislocation density is u = v x . This model has been recently studied by Biler, Karch and Monneau in [4], where they prove that the problem enjoys the properties of uniqueness and comparison of viscosity solutions. The relation between u and v is very interesting and will be used by us in the final sections.Another possible generalization of the (CV) model is ∂ t u = ∇ · (u∇p), p = (−∆) −s (|u| m−2 u), that has been investigated by Biler, Imbert and Karch in [2,3]. They prove the existence of weak solutions and they find explicit self-similar solutions with compact support for all m > 1. The finite speed of propagation for every weak solution has been done in [22].The second nonlocal version of the classical PME is the model u t = −(−∆) s u m , m > 0,

We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general theory. We introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one, thus providing us with a number of new solutions, as well as interesting connections. Special attention is paid to the property of finite propagation.