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 $$(\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\mathrm{\Delta })u\le A{u}^p+Bu$$ with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $L^{\mathrm{\infty }}$ estimates assuming $$\mathrm{\parallel }u(\cdot ,0){\mathrm{\parallel }}_{M^{q,\frac{2q}{p-1}}}+\underset{0\le t<T}{\sup }\mathrm{\parallel }u(\cdot ,t){\mathrm{\parallel }}_{L^s}<\delta ,$$ where $1<q\le q_c:=\frac{n(p-1)}{2}$ and $1\le s\le q_c$. Assuming also a bound on $\mathrm{\parallel }u(\cdot ,0){\mathrm{\parallel }}_{M^{q^{\mathrm{\prime }},{\lambda }^{\mathrm{\prime }}}}$, where either $q^{\mathrm{\prime }}>q$ or ${\lambda }^{\mathrm{\prime }}<\frac{2q}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions.