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Hamilton-Souplet-Zhang’s gradient estimates for two weighted nonlinear parabolic equations

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Abstract

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is

$${u_t} = {\Delta _f}u + au\log u + bu$$

with a, b two real constants, and another is

$${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$

with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.

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Authors and Affiliations

  1. Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

    Bing-qing Ma & Guang-yue Huang

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  1. Bing-qing Ma
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  2. Guang-yue Huang
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Correspondence to Bing-qing Ma.

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Supported by NSFC(11371018, 11401179, 11671121).

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Ma, Bq., Huang, Gy. Hamilton-Souplet-Zhang’s gradient estimates for two weighted nonlinear parabolic equations. Appl. Math. J. Chin. Univ. 32, 353–364 (2017). https://doi.org/10.1007/s11766-017-3500-x

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  • DOI: https://doi.org/10.1007/s11766-017-3500-x

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