TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS

doi:10.1007/s10473-022-0424-1

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1690-1700.doi: 10.1007/s10473-022-0424-1

• 论文 • 上一篇

TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS

吴加勇

Department of Mathematics, Shanghai University, Shanghai, 200444, China

收稿日期:2021-07-16 修回日期:2021-10-17 出版日期:2022-08-25 发布日期:2022-08-23
通讯作者: Jiayong WU,E-mail:wujiayong@shu.edu.cn E-mail:wujiayong@shu.edu.cn
基金资助:
This work was partially supported by the National Natural Science Foundation of China (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800).

TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS

Jiayong WU

Department of Mathematics, Shanghai University, Shanghai, 200444, China

Received:2021-07-16 Revised:2021-10-17 Online:2022-08-25 Published:2022-08-23
Contact: Jiayong WU,E-mail:wujiayong@shu.edu.cn E-mail:wujiayong@shu.edu.cn
Supported by:
This work was partially supported by the National Natural Science Foundation of China (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800).

摘要/Abstract

摘要： On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

关键词: Gradient shrinking Ricci soliton, heat equation, time analyticity

Abstract: On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

Key words: Gradient shrinking Ricci soliton, heat equation, time analyticity

中图分类号:

53C21

吴加勇. TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS[J]. 数学物理学报(英文版), 2022, 42(4): 1690-1700.

Jiayong WU. TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1690-1700.

参考文献

[1] Widder D V, Analytic solutions of the heat equation. Duke Math J, 1962, 29:497-503
[2] Zhang Q S. A note on time analyticity for ancient solutions of the heat equation. Proc Amer Math Soc, 2020, 148(4):1665-1670
[3] Dong H J, Zhang Q S. Time analyticity for the heat equation and Navier-Stokes equations. J Funct Anal, 2020, 279(4):108563, 15pp
[4] Masuda K, On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation. Proc Japan Acad, 1967, 43:827-832
[5] Kinderlehrer D, Nirenberg L. Analyticity at the boundary of solutions of nonlinear second-order parabolic equations. Comm Pure Appl Math, 1978, 31(3):283-338
[6] Komatsu G. Global analyticity up to the boundary of solutions of the Navier-Stokes equation. Comm Pure Appl Math, 1980, 33(4):545-566
[7] Giga Y. Time and spatial analyticity of solutions of the Navier-Stokes equations. Comm Partial Differential Equations, 1983, 8(8):929-948
[8] Escauriaza L, Montaner S, Zhang C. Analyticity of solutions to parabolic evolutions and applications. SIAM J Math Anal, 2017, 49(5):4064-4092
[9] Han F W, Hua B B, Wang L L. Time analyticity of solutions to the heat equation on graphs. Proc Amer Math Soc, 2021, 149(6):2279-2290
[10] Hamilton R. The formation of singularities in the Ricci flow, Surveys in Differential Geometry. Boston:International Press, 1995, 2:7-136
[11] Perelman G. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
[12] Perelman G. Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109
[13] Perelman G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245
[14] Cao H D. Recent progress on Ricci solitons, Recent Advances in Geometric Analysis//Lee Y -I, Lin C -S, Tsui M -P, Advanced Lectures in Mathematics (ALM). Somerville:International Press, 2010, 11:1-38
[15] Carrillo J, Ni L. Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm Anal Geom, 2009, 17(4):721-753
[16] Chen B L. Strong uniqueness of the Ricci flow. J Diff Geom, 2009, 82(2):363-382
[17] Pigola S, Rimoldi M, Setti A G. Remarks on non-compact gradient Ricci solitons. Math Z, 2011, 268(3/8):777-790
[18] Cao H D, Zhou D T. On complete gradient shrinking Ricci solitons. J Diff Geom, 2010, 85(2):175-186
[19] Chow B, Chu S C, Glickenstein D, et al. The Ricci flow:techniques and applications, part IV:long-time solutions and related topics. Mathematical Surveys and Monographs. Vol 206. Providence, RI:American Mathematical Society, 2015
[20] Munteanu O. The volume growth of complete gradient shrinking Ricci solitons. arXiv:0904.0798v2
[21] Munteanu O, Wang J P, Geometry of manifolds with densities. Adv Math, 2014, 259:269-305
[22] Haslhofer R, Müller R. A compactness theorem for complete Ricci shrinkers. Geom Funct Anal, 2011, 21(5):1091-1116
[23] Li Y, Wang B. Heat kernel on Ricci shrinkers. Calc Var Partial Differential Equations, 2020, 59(6):Art 194
[24] Zhang Q S. Sobolev inequalities, heat kernels under Ricci flow, and the Poincaré conjecture. Boca Raton, FL:CRC Press, 2011
[25] Wu J Y, Wu P. Heat kernels on smooth metric measure spaces with nonnegative curvature. Math Ann, 2015, 362(3/8):717-742

TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS

TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 12

Metrics

本文评价

推荐阅读 10

[1]	王军, 陈振龙. HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS[J]. 数学物理学报(英文版), 2022, 42(2): 653-670.
[2]	余成杰, 赵菲菲. A NOTE ON LI-YAU-TYPE GRADIENT ESTIMATE[J]. 数学物理学报(英文版), 2019, 39(4): 1185-1194.
[3]	李贺宇, 陈夏. PRECISE MOMENT ASYMPTOTICS FOR THE STOCHASTIC HEAT EQUATION OF A TIME-DERIVATIVE GAUSSIAN NOISE[J]. 数学物理学报(英文版), 2019, 39(3): 629-644.
[4]	陈乐, Kunwoo KIM. NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY[J]. 数学物理学报(英文版), 2019, 39(3): 645-668.
[5]	胡耀忠, Khoa LÊ. JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL[J]. 数学物理学报(英文版), 2019, 39(3): 764-780.
[6]	胡耀忠, 刘阳辉, Samy TINDEL. ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE[J]. 数学物理学报(英文版), 2019, 39(3): 669-690.
[7]	胡耀忠. SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(3): 874-914.
[8]	Abderrahmane YOUKANA, Salim A. MESSAOUDI, Aissa GUESMIA. A GENERAL DECAY AND OPTIMAL DECAY RESULT IN A HEAT SYSTEM WITH A VISCOELASTIC TERM[J]. 数学物理学报(英文版), 2019, 39(2): 618-626.
[9]	王宇钊, 杨杰, 陈文艺. GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES[J]. 数学物理学报(英文版), 2013, 33(4): 963-974.
[10]	谭忠, 吴国春. ON THE HEAT FLOW EQUATION OF SURFACES OF CONSTANT MEAN CURVATURE IN HIGHER DIMENSIONS[J]. 数学物理学报(英文版), 2011, 31(5): 1741-1748.
[11]	林支桂. THE BLOW-UP PROPERLIES OF SOLUTIONS TO SEMILINEAR HEAT EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 1998, 18(3): 315-325.
[12]	叶兴德, 江金生. A NEW PSEUDOSPECTRAL APPROXIMATION FOR THE FOWARD-BACKWARD HEAT EQUATION[J]. 数学物理学报(英文版), 1996, 16(2): 121-128.