Global existence for the nonlinear heat equation on riemannian manifolds with negative sectional curvature

We address local existence, blow-up and global existence of mild solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature. We deal with a power nonlinearity multiplied by a time-dependent positive function h(t), and initial conditions u0 08Lp(M). 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, for any power nonlinearity, if h 611 we have global existence for small initial data, whereas if h(t)=e\u3b1t a Fujita type phenomenon prevails varying the parameter \u3b1>0