We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows.We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

We study weighted porous media equations on domains Ω ⊆ R N , either with Dirichlet or with Neumann homogeneous boundary conditions when Ω = R N . Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, L q 0 -L ̺ smoothing effects (1 ≤ q0 < ̺ < ∞) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain L q 0 -L ̺ smoothing effects and suitable weighted Poincaré-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case Ω = R N when the corresponding weight makes its measure finite, so that solutions converge to their weighted mean value instead than to zero. Examples are given in terms of wide classes of weights.

Abstract. We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at infinity too fast. This proves in particular that the growth rate of the data we consider is optimal for existence. Pointwise blow-up is also shown for a particular class of manifolds and of initial data.

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified. Symmétrie des fonctions extrémales pour des inégalités de Caffarelli-Kohn-Nirenberg sous-critiquesNous utilisons le formalisme des entropies de Rényi pour établir le domaine de symétrie des fonctions extrémales dans une famille d'inégalités de Caffarelli-Kohn-Nirenberg sous-critiques. Par fonctions extrémales, il faut comprendre des fonctions qui réalisent le cas d'égalité dans les inégalités écrites avec des constantes optimales. La méthode étend des résultats récents sur les inégalités de Caffarelli-Kohn-Nirenberg critiques. En utilisant une heuristique donnée par une équation de diffusion non-linéaire, nous donnons une preuve variationnelle d'un résultat de symétrie, grâce à un théorème de rigidité : dans la région de symétrie, tous les points critiques positifs sont à symétrie radiale et sont par conséquent égaux à l'unique point critique radial, positif, à une multiplication par une constante et à un changement d'échelle près. Ce résultat est optimal. La condition sur les paramètres est en effet complémentaire de celle qui définit la région dans laquelle il y a brisure de symétrie du fait de l'instabilité linéaire des fonctions radiales optimales. Comparé au cas critique, le domaine sous-critique nécessite de nouveaux outils. L'information de Fisher doit être remplacée par l'entropie de Rényi, et comme certaines invariances sont perdues, les estimations basées sur la transformation d'Emden-Fowler doivent être modifiées.

We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space R d . For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (powerlike). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in Grillo et al. (Discret Contin Dyn Syst 35:5927-5962, 2015), where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in Vázquez (J Eur Math Soc 16:769-803, 2014). Mathematics Subject Classification