Hamilton’s gradient estimate for fast diffusion equations under geometric flow

Ghodratallah Fasihi-Ramandi; Ghodratallah Fasihi-Ramandi

doi:10.3934/math.2019.3.497

AIMS Mathematics

2019, Volume 4, Issue 3: 497-505. doi: 10.3934/math.2019.3.497

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Research article

Hamilton’s gradient estimate for fast diffusion equations under geometric flow

Ghodratallah Fasihi-Ramandi ^,

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Received: 08 January 2019 Accepted: 08 May 2019 Published: 15 May 2019
MSC : 53C21, 53C44, 58J35

Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.
- fast diffusion equation,
- Ricci flow,
- Hamilton inequality,
- gradient estimates
Citation: Ghodratallah Fasihi-Ramandi. Hamilton’s gradient estimate for fast diffusion equations under geometric flow[J]. AIMS Mathematics, 2019, 4(3): 497-505. doi: 10.3934/math.2019.3.497

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Abstract

Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.

References

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