Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature

Abstract

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

Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds

“…We obtain global existence also for h(t) = e σt (σ > 0) and p > 1 + σ λ1(M) . On the other hand, for p < 1 + σ λ1(M) , there is finite time blow-up, by results in [19]. Observe that, in order to deal with special functions h, we prove that there exist λ ∈ (0, λ 1 ] and a weak bounded supersolution of (1.3) for such λ.…”

“…• Assume (A 0 ) and (A 1 ). For h ≡ 1, from the results in [20] we get global existence, if u 0 L n 2 (p−1) (M) or u 0 L p (M) is small enough. This assumption is clearly different in character form those made in Theorems 3.1, 3.2, 3.4, 3.5.…”

“…Note that in general the solution of problem (1.1) is not unique in L ∞ (M × (0, T )). Conditions, related to M , that guarantee uniqueness for problem (1.1) are established in [19] (see also e.g. [9] for linear equations).…”

“…[9] for linear equations). For further references, we recall that in [19] it is proved that if u 0 ≥ 0, u 0 ≡ 0, and • Let γ > 0. For any λ > 0, assumption (3.1) is satisfied.…”

“…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].…”