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 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

We study existence and nonexistence of patterns on Riemannian manifolds, depending on the Ricci curvature of the manifold and suitable assumptions on the boundary. In the case of surfaces of revolutions in R(3), necessary and sufficient conditions for existence of patterns are given. (C) 2011 Elsevier Inc. All rights reserved

We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density ρ>0. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according to the behavior of ρ at infinity. © 2013 Elsevier B.V. All rights reserved

On Riemannian manifolds with negative sectional curvature, we study finite time blow-up and global existence of solutions to semilinear parabolic equations, where the power nonlinearity is multiplied by a time-dependent positive function h(t). We show that depending on the behavior at infinity of h, either every solution blows up in finite time, or a global solution exists, if the initial datum is small enough. In particular, if h≡1 we have global existence for small initial data, whereas for h(t)=eαt a Fujita-type phenomenon appears for certain values of α>0. A key role will be played by the infimum of the L2-spectrum of the operator -δ on M. © 2011 Elsevier Inc