In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.

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 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.

We study the general nonlinear diffusion equation u t = ∇ · (u m−1 ∇(−∆) −s u) that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters m > 1 and 0 < s < 1, we assume that the solutions are non-negative and the problem is posed in the whole space. In this paper we prove existence of weak solutions for all integrable initial data u 0 ≥ 0 and for all exponents m > 1 by developing a new approximation method that allows to treat the range m ≥ 3, that could not be covered by previous works. We also extend the class of initial data to include any non-negative measure µ with finite mass. In passing from bounded initial data to measure data we make strong use of an L 1 -L ∞ smoothing effect and other functional estimates. Finite speed of propagation is established for all m ≥ 2, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for m < 2.