The semilinear heat inequality with Morrey initial data on Riemannian manifolds

Abstract The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) u \le A u^p + B u \nonumber \end{aligned}$$ ∂ ∂ t - Δ u ≤ A u p + B u with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $$L^\infty $$ L ∞ estimates assuming $$\Vert u(\cdot ,0)\Vert _{M^{q, \frac{2q}{p-1}}} + \sup _{0 \le t< T} \Vert u(\cdot , t) \Vert _{L^s} < \delta ,$$ ‖ u ( · , 0 ) ‖ M q , 2 q p - 1 + sup 0 ≤ t < T ‖ u ( · , t ) ‖ L s < δ , where $$1 < q \le q_c:= \frac{n(p-1)}{2}$$ 1 < q ≤ q c : = n ( p - 1 ) 2 and $$1 \le s \le q_c$$ 1 ≤ s ≤ q c . Assuming also a bound on $$\Vert u(\cdot , 0)\Vert _{M^{q', \lambda '}}$$ ‖ u ( · , 0 ) ‖ M q ′