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.
Categories: math.AP math.DG