The semilinear heat inequality with Morrey initial data on Riemannian manifolds

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.