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

Final version, appeared in J. Evol. Equ. 25 pages