Solvability of a semilinear heat equation on Riemannian manifolds

TL;DR

This paper investigates the conditions under which the semilinear heat equation with measure initial data is solvable on Riemannian manifolds, providing sharp criteria for local-in-time solutions.

Contribution

It establishes precise conditions for local-in-time solvability of the semilinear heat equation on Riemannian manifolds with specific geometric bounds.

Findings

Sharp solvability conditions derived for complete, connected manifolds

Criteria depend on geometric properties like injectivity radius and curvature

Results applicable to initial data as nonnegative Radon measures

Abstract

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.