Mathematics > Analysis of PDEs
arXiv:2412.21029 (math)
[Submitted on 30 Dec 2024 (v1), last revised 24 Jun 2026 (this version, v2)]
Abstract:The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - \Delta\right) u \leq A u^p + B u $$ 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 $\frac{\lambda'}{2q'} < \frac{1}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions.
Submission history
From: Anuk Dayaprema [view email]
[v1]
Mon, 30 Dec 2024 15:55:58 UTC (36 KB)
[v2]
Wed, 24 Jun 2026 03:10:10 UTC (37 KB)
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