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L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita's phenomenon. To have the same situation as for the Cauchy problem in R-N, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles. (C) 2011 Published by Elsevier Inc

Abstract.In this paper we study the first initial-boundary value problem for u, = Au + up in conical domains D = (0,oo) x Í2 c RN where Í2 C SN~l is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case D = RN .Let X = -y-where y_ is the negative root of y(y + N -2) = wx and where wx is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on Í2 . We prove: If 1 < p < 1 + 2/(2 + X), there are no nontrivial global solutions. For 1 < p < oo , there are L00 data of arbitrarily small norm, decaying exponentially fast at r = oo , for which the solution is not global.We show that if D is the exterior of a bounded region, there are no global, nontrivial, positive solutions if 1 < p < 1 + 2/N and that there are such if p > 1 +2/N . We obtain some related results for u, = Au+ \x\"up in the cone.