The semilinear heat inequality with Morrey initial data on Riemannian manifolds

TL;DR

This paper derives estimates for solutions of a semilinear heat inequality with Morrey space initial data on Riemannian manifolds, providing bounds that depend on initial data norms and applicable to geometric flows.

Contribution

It introduces new $L^$ estimates for solutions with Morrey initial data on manifolds, extending previous results to a geometric setting with bounded geometry.

Findings

Established $L^$ bounds under small Morrey norm initial data.

Derived improved estimates near initial time with additional bounds.

Applied results to geometric flows in higher dimensions.

Abstract

The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - \Delta\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $L^\infty$ estimates assuming $$\|u(\cdot,0)\|_{M^{q, \frac{2q}{p-1}}} + \sup_{0 \leq t < T} \|u(\cdot, t) \|_{L^s} < \delta,$$ where $1 < q \leq q_c := \frac{n(p-1)}{2}$ and $1 \leq s \leq q_c$. Assuming also a bound on $\|u(\cdot, 0)\|_{M^{q', \lambda'}}$, where either $q' > q$ or $\lambda' < \frac{2q}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions.