  We study the solvability of the initial value problem for the semilinear heat
equation $u_t-\Delta u=u^p$ in a Riemannian manifold $M$ with a nonnegative
Radon measure $\mu$ on $M$ as initial data. We give sharp conditions on the
local-in-time solvability of the problem for complete and connected $M$ with
positive injectivity radius and bounded sectional curvature.
