Solvability of a semilinear heat equation on Riemannian manifolds

We study the solvability of the initial value problem for the semilinear heat equation $u_t-\mathrm{\Delta }u=u^p$ 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.