대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제60권6호
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  • Pages.1673-1685
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  • 2023
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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TIME ANALYTICITY FOR THE HEAT EQUATION UNDER BAKRY-ÉMERY RICCI CURVATURE CONDITION

  • Ling Wu (School of Mathematical Sciences and Shanghai Key Laboratory of PMMP East China Normal University)
  • 투고 : 2022.12.07
  • 심사 : 2023.07.14
  • 발행 : 2023.11.30
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⟨ 이전 논문 다음 논문 ⟩

초록

Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain Lp spaces with p ∈ [2, +∞) and prove its analyticity with respect to time.

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과제정보

The author would like to thank Professor Meng Zhu for very helpful discussions and invaluable suggestions. Thanks also goes to the referee for checking the paper carefully and making useful suggestions. Research is partially supported by NSFC Grant No. 11971168, Shanghai Science and Technology Innovation Program Basic Research Project STCSM 20JC1412900, and Science and Technology Commission of Shanghai Municipality (STCSM) No. 22DZ2229014.

참고문헌

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