3,304 research outputs found
Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations
Full text linkWe investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for integrable initial data was established in Bonforte, Grillo, and Vazquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, in the Euclidean space, it is known from Herrero and Pierre [Trans. Amer. Math. Soc. 291 (1985), pp. 145–158], that the Cauchy problem associated with the FDE is well posed for initial data that are merely locally integrable. We establish here that such data still give rise to
global solutions on general manifolds. If, moreover, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to minus infinity at
spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, assuming in addition that the initial
datum is locally square integrable and nonnegative, a minimal solution is shown to exist, and we establish uniqueness of purely (nonnegative) distributional solutions, a fact
that to our knowledge was not known before even in the Euclidean space. The required curvature bound is sharp, since on model manifolds it is equivalent to stochastic completeness, and it was shown in Grillo, Ishige, and Muratori [J. Math. Pures Appl. (9) 139 (2020), pp. 63–82] that uniqueness for the FDE fails even in the class of bounded solutions when stochastic completeness does not hold. A crucial ingredient of the uniqueness result is the proof of nonexistence of nonnegative, nontrivial distributional subsolutions to certain
semilinear elliptic equations with power nonlinearities, of independent interest
Fractional porous media equations: existence and uniqueness of weak solutions with measure data
Full text linkWe 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 Rd. 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 (power-like). 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)
Five Stars and a Cricket. Beppe Grillo Shakes Italian Politics
No full textThe article focuses on a new political player: the Five Star Movement led by the comedian Beppe Grillo. The party lies at the junction between different organisational models and conceptions of democracy: it combines an online and an offline presence; it has ‘horizontal’ structural elements, but a top-down decision-making process; it is positioned ‘beyond’ ideologies, while its electorate comes from various political families. The work considers the history, message, leader, organisation and electoral base of the movement, as well as the political opportunity structure that facilitated its growth in 2012 and the challenges it faces in the delicate phase of institutionalisation
Dos raros en Paris. Felisberto Hernandez y Susana Soca
No full textEsiste una amplissima bibliografia sui viaggi in Europa, e principalmente a Parigi, di scrittori latinoamericani, che cercavano nella Ville Lumière la consacrazione nel 'Primo Mondo' e un ritorno glorioso nella nazione di appartenenza. Nel saggio si analizzano la natura, gli incontri, la produzione letteraria e gli eventi del viaggio di Felisberto Hernández a Parigi in relazione non a Jules Supervielle, il grande poeta franco-uruguaiano che fu il mecenate del viaggio di Felisberto e su cui già si è scritto molto, ma a Susana Soca, raffinata poetessa e intellettuale uruguaiana da sempre promotrice di fecondi rapporti culturali e artistici tra i due paesi, tra cui la rivista "La Licorne" e "Entregas de La Licorne", pubblicata la prima a Parigi e la seconda a Montevideo. Al di là dei rapporti personali, è sommamente interessante ricostruire le relazioni, gli stimoli e gli arricchimenti culturali scaturiti dai due anni trascorsi da Felisberto a Parigi, di cui parla diffusamente nell'epistolario e nelle opere successive
Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups
No full textThe porous medium equation with large initial data on negatively curved Riemannian manifolds
Full text linkWe 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. In particular, the pressure at infinity can grow at most linearly on the hyperbolic space, and quadratically both on the Euclidean space and on a class of manifolds whose Ricci curvature vanishes sufficiently fast at infinity.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. On montre l'existence et l'unicité de solutions trà ̈s faibles du problà ̈me de Cauchy pour l'équation des milieux poreux sur variétés de Cartan-Hadamard qui satisfont des bornes inférieures sur la courbure de Ricci, avec des données initiales qui peuvent croître à l'infini à un taux contrà ́lé, qui dépend de manià ̈re cruciale des bornes de courbure. En particulier, la pression à l'infini peut croître au plus linéairement sur l'espace hyperbolique, et quadratiquement sur l'espace euclidien ainsi que dans une classe de variétés dont la courbure de Ricci tend vers zéro de manià ̈re suffisamment rapide à l'infini.Les conditions de courbure qu'on impose sont optimales en ce qui concerne l'unicité, dans la mesure oÃ1, si elles ne sont pas satisfaites, en général il peut y avoir une infinité de solutions du problà ̈me de Cauchy, mÃame avec des données initiales bornées. En plus, en imposant des bornes supérieures sur les courbures sectionnelles qui correspondent à celles inférieures, on fournit une estimation précise du temps maximal d'existence, et on démontre qu'en général les solutions n'existent pas si la donnée initiale croît trop vite à l'infini. Cela prouve, en particulier, que le taux de croissance de la donnée initiale qu'on considà ̈re est optimal pour l'existence. Finalement, on montre un résultat d'explosion ponctuelle pour une classe particulià ̈re de variétés et de données initiales
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