In this paper, we mainly consider the solvability of the initial value problem for the semilinear parabolic equation \(u_t-\triangle u+u=u^p\) in a Riemannian manifold M with a nonnegative Radon measure \(\mu \) on M as initial data. When M is a N-dimensional connected and complete Riemannian manifold without boundary, and has bounded sectional curvature and positive injectivity radius, Takahashi and Yamamoto (J Evol Equ 23:55, 2023) have already studied the solvability of semilinear heat equation \(u_t-\triangle u=u^p\) . We use a substitution \(u=e^{-t}v\) , then we can transform the semilinear parabolic equation \(u_t-\triangle u+u=u^p\) to the semilinear heat equation \(v_t-\triangle v=e^{-(p-1)t}v^p\) . Finally we get the sharp conditions and necessary conditions on the local-in-time solvability of the semilinear parabolic equation by using the properties of the heat kernel.
