The Fujita exponent for the Cauchy problem in the hyperbolic space

Abstract

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

Blow-up and global existence for the porous medium equation with reaction on a class of Cartan–Hadamard manifolds

“…The behaviour of solutions to problem (1.1) is therefore determined by competing phenomena: the diffusion pattern associated to −∆, the reaction due to the power source, and the (slow, but faster than in the Euclidean case) diffusion properties of the porous medium equation u t = ∆(u m ). In fact, in the case of linear diffusion (m = 1) it is known (see [2], [43], [44], [32]) that, when M = H N , for all p > 1 and sufficiently small nonnegative data there exists a global in time solution. The situation is different in R N : indeed, blowup occurs for all nontrivial nonnegative data when p ≤ 1 + 2/N , while global existence prevails for p > 1 + 2/N (for more specific results, see e.g.…”

Nonexistence for hyperbolic problems on Riemannian manifolds1

“…We use the following result established in [3]. 2]. Then there exists a family of functions {φ R } R≥1 ⊂ C ∞ c (M ) with the following properties:…”

“…Levine [4] and the references therein. More recently, an extension of this kind of achievements when the Euclidean space is replaced by noncompact Riemannian manifolds was obtained in [2,12,14,17], under suitable geometric hypotheses. Similar results for elliptic equations on noncompact Riemannian manifolds have been also investigated (see, e.g., [7], [8], [11]).…”

Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds

“…On the contrary, if p > 1 + 2 n and the initial datum u 0 is small enough, then global solutions exist. Problem (1.1) with M the hyperbolic space H n has been addressed in [1]. It is shown that if h(t) ≡ 1 (t ≥ 0) or…”

“…However, such hypotheses cannot be satisfied on Riemannian manifolds with strictly negative sectional curvatures. On the other hand, in [19] the results established in [1] have been generalized to Cartan-Hadamard manifolds M with sectional curvatures bounded above by a negative constant. Some global existence results for mild solutions belonging to C([0, T ); L p (M )) have been established in [20], by using general results in semigroup theory stated in [22].…”