Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds

Abstract

We are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds.

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

“…We consider only two special subclass of the problem. If γ > 0, then there exists a positive bounded super-solution to −∆ M φ = λ 1 (M )φ which follows from the work of F. Punzo [27] and we can get sharp bounds on the exponent such that Fujita phenomena holds. On the other hand if we allow γ ≥ 0, then we can prove some partial results stated below.…”

“…• General Cartan-Hadamard manifolds. In addition to the hyperbolic space, following the ideas of F. Punzo in [27], we can extend the analogous results in the case of a Cartan-Hadamard manifold whose sectional curvature is bounded by a negative constant. It is important to note that, except possibly at the borderline case, many of the results of this article continue to hold true for Cartan-Hadamard manifolds with a pole, under the curvature bound K R ≤ −c, where K R being the sectional curvature in the radial direction (see Theorem 3.1 and Theorem 3.2).…”

“…The main idea rests on the existence of positive bounded super-solution. In [27], it was shown using Lemma 3.1 that generalised eigenvalue problem admits positive bounded super-solution under the above mentioned assumptions on the manifold. For the hyperbolic space there exists a bounded ground state which suffices the purpose.…”

“…For the hyperbolic space there exists a bounded ground state which suffices the purpose. If we consider, ψ(r) = 1 κ sinh(κr) then the model manifold has constant sectional curvature −κ 2 , the bottom of the spectrum is given by (n−1) 2 comparison principle of weak solutions to (1.1) and a positive bounded weak super solution (see [27]) to the eigenvalue problem :…”

“…(3) If p < p * F (µ), then all non-negative solutions to (1.6) blow-up in finite time. Recently, considerable efforts have been made by several authors (see [20,27,28,38] and the references in therein) to extend these results to Riemannian manifolds M, satisfying certain curvature bounds. More generally, the question of Fujita phenomena for Porous medium equation has also drawn significant development in this direction.…”