The semilinear heat inequality with Morrey initial data on Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915085266452480 |
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| author | Dayaprema, Anuk |
| author_facet | Dayaprema, Anuk |
| contents | The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - Δ\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $L^\infty$ estimates assuming $$\|u(\cdot,0)\|_{M^{q, \frac{2q}{p-1}}} + \sup_{0 \leq t < T} \|u(\cdot, t) \|_{L^s} < δ,$$ where $1 < q \leq q_c := \frac{n(p-1)}{2}$ and $1 \leq s \leq q_c$. Assuming also a bound on $\|u(\cdot, 0)\|_{M^{q', λ'}}$, where either $q' > q$ or $λ' < \frac{2q}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_21029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The semilinear heat inequality with Morrey initial data on Riemannian manifolds Dayaprema, Anuk Analysis of PDEs Differential Geometry The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - Δ\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $L^\infty$ estimates assuming $$\|u(\cdot,0)\|_{M^{q, \frac{2q}{p-1}}} + \sup_{0 \leq t < T} \|u(\cdot, t) \|_{L^s} < δ,$$ where $1 < q \leq q_c := \frac{n(p-1)}{2}$ and $1 \leq s \leq q_c$. Assuming also a bound on $\|u(\cdot, 0)\|_{M^{q', λ'}}$, where either $q' > q$ or $λ' < \frac{2q}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions. |
| title | The semilinear heat inequality with Morrey initial data on Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2412.21029 |