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

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 ) = e α t a Fujita-type phenomenon appears for certain values of α > 0 . A key role will be played by the infimum of the L 2 -spectrum of the operator −Δ on M.