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