ON THE HEAT FLOW OF EQUATION OF H-SURFACE

doi:10.1007/s10473-019-0516-8

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1397-1405.doi: 10.1007/s10473-019-0516-8

ON THE HEAT FLOW OF EQUATION OF H-SURFACE

吴国春¹, 谭忠², 许建开³

1. Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China;
2. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen 361005, China;
3. College of Information Science and Technology, Hunan Agriculture University, Changsha 410128, China

收稿日期:2017-09-25 修回日期:2019-03-04 出版日期:2019-10-25 发布日期:2019-11-11
通讯作者: Zhong TAN E-mail:ztan85@163.com
作者简介:Guochun WU,E-mail:guochunwu@126.com;Jiankai XU,E-mail:jiankaixu@126.com
基金资助:
Guochun Wu's research was in part supported by National Natural Science Foundation of China (11701193, 11671086), Natural Science Foundation of Fujian Province (2018J05005), Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan (2017ZT012). Zhong Tan's research was in part supported by National Natural Science Foundation of China (11271305, 11531010). Jiankai Xu's research was in part supported by National Natural Science Foundation (11671086, 11871208), Natural Science Foundation of Hunan Province (2018JJ2159).

ON THE HEAT FLOW OF EQUATION OF H-SURFACE

Guochun WU¹, Zhong TAN², Jiankai XU³

1. Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China;
2. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen 361005, China;
3. College of Information Science and Technology, Hunan Agriculture University, Changsha 410128, China

Received:2017-09-25 Revised:2019-03-04 Online:2019-10-25 Published:2019-11-11
Contact: Zhong TAN E-mail:ztan85@163.com
Supported by:
Guochun Wu's research was in part supported by National Natural Science Foundation of China (11701193, 11671086), Natural Science Foundation of Fujian Province (2018J05005), Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan (2017ZT012). Zhong Tan's research was in part supported by National Natural Science Foundation of China (11271305, 11531010). Jiankai Xu's research was in part supported by National Natural Science Foundation (11671086, 11871208), Natural Science Foundation of Hunan Province (2018JJ2159).

摘要/Abstract

摘要： We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M₂ and "unstable set" M₁, we show that there exists a unique global solution provided the initial data belong to M₂ and the global solution converges to zero in H¹ exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M₁.

关键词: H-surface, non-zero Dirichlet boundary, singularity, global solutions

Abstract: We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M₂ and "unstable set" M₁, we show that there exists a unique global solution provided the initial data belong to M₂ and the global solution converges to zero in H¹ exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M₁.

Key words: H-surface, non-zero Dirichlet boundary, singularity, global solutions

中图分类号:

35J20

吴国春, 谭忠, 许建开. ON THE HEAT FLOW OF EQUATION OF H-SURFACE[J]. 数学物理学报(英文版), 2019, 39(5): 1397-1405.

Guochun WU, Zhong TAN, Jiankai XU. ON THE HEAT FLOW OF EQUATION OF H-SURFACE[J]. Acta mathematica scientia,Series B, 2019, 39(5): 1397-1405.

参考文献

[1] Bethuel F, Caldiroli P, Guida M. Parametric Surfaces with Prescribed Mean Curvature. Rend Sem Mat Univ Torino, 2002, 60(4):175-231
[2] Bögelein V, Duzaar F, Scheven C. Weak solutions to the heat flow for surfaces of prescribed mean curvature. Trans Amer Math Soc, 2013, 365(9):4633-4677
[3] Bögelein V, Duzaar F, Scheven C. Short-time regularity for the H-surface flow. Int Math Res Not IMRN, 2015
[4] Brezis H, Coron J M. Multiple Solutions of H-systems and Rellich's conjecture. Comm Pure Appl Math, 1984, 37(2):147-187
[5] Brezis H, Coron J M. Convergence of solutions of H-systems or how to blow bubbles. Arch Rat Mech Anal, 1985, 89(1):21-56
[6] Caldiroli P, Musina R. The Dirichlet problem for H-systems with small boundary data:Blowup phenomena and nonexistence results. Arch Rat Mech Anal, 2006, 181(1):142-183
[7] Chang K C. Heat flow and boundary value problem for harmonic maps. Annales de l'institut Henri Poincaré C, Analyse non linéaire, 1989, 6(5):363-395
[8] Chang K C, Liu J Q. Heat flow for the minimal surface with Plateau boundary condition. Acta Mathematica Sinica (English series), 2003, 19(1):1-28
[9] Chang K C, Liu J Q. Another approach to the heat flow for Plateau problem. J Differential Equations, 2003, 189(1):46-70
[10] Chang K C, Liu J Q. An evolution of minimal surfaces with Plateau condition. Calc Var, 2004, 19(2):117-163
[11] Chang K C, Liu J Q. Boundary flow for the minimal surfaces in Rn with Plateau boundary condition. Proc Roy Soc Edinburgh Sect A, 2005, 135(3):537-562
[12] Chen Y, Levine S. The existence of the heat flow for H-systems. Disc Cont Dyna Syst, 2002, 8(1):219-236
[13] Hildebrant S. On the Plateau problem for surfaces of constant mean curvature. Comm Pure Appl Math, 1970, 23(1):97-114
[14] Huang T, Tan Z, Wang C Y. On the heat flow of equation of surfaces of constant mean curvature. Manuscripta Math, 2011, 134(1/2):259-271
[15] Levine H A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_t=-Au + F(u). Arch Rat Mech Anal, 1973, 51(5):371-386
[16] Rey O. Heat flow for the equation of surfaces with prescribed mean curvature. Math Ann, 1991, 297(1):123-146
[17] Struwe M. The existence of surfaces of constant mean curvature with free boundaries. Acta Math, 1988, 160(1):19-64
[18] Tan Z. Global solutions and blowup of semilinear heat equation with critical sobolev exponent. Commun Partial Differential Equations, 2001, 26(3/4):717-741
[19] Tan Z. The reaction-diffusion equation with Lewis function and critical Sobolev exponent. J Math Anal Appl, 2002, 272(2):480-495
[20] Tan Z. Asymptotic behavior and blowup of some degenerate parabolic equation with critical Sobolev exponent. Commun Appl Anal, 2004, 8(1):67-85
[21] Wente H C. An existence theorem for surfaces of constant mean curvature. J Math Anal Appl, 1969, 26(2):318-344

ON THE HEAT FLOW OF EQUATION OF H-SURFACE

ON THE HEAT FLOW OF EQUATION OF H-SURFACE

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 13

Metrics

本文评价

推荐阅读 0

[1]	Abdul-Majeed AL-IZERI, Khalid LATRACH. NONLINEAR SEMIGROUP APPROACH TO TRANSPORT EQUATIONS WITH DELAYED NEUTRONS[J]. 数学物理学报(英文版), 2018, 38(6): 1637-1654.
[2]	牛海萍, 王术. SOLUTIONS TO QUASILINEAR HYPERBOLIC CONSERVATION LAWS WITH INITIAL DISCONTINUITIES[J]. 数学物理学报(英文版), 2018, 38(1): 203-219.
[3]	杨东成. THE VLASOV-MAXWELL-FOKKER-PLANCK SYSTEM WITH RELATIVISTIC TRANSPORT IN THE WHOLE SPACE[J]. 数学物理学报(英文版), 2017, 37(5): 1237-1261.
[4]	Abdul-Majeed AL-IZERI, Khalid LATRACH. WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH[J]. 数学物理学报(英文版), 2016, 36(5): 1225-1244.
[5]	王路生, 肖清华, 熊林杰, 赵会江. THE VLASOV-POISSON-BOLTZMANN SYSTEM NEAR MAXWELLIANS FOR LONG-RANGE INTERACTIONS[J]. 数学物理学报(英文版), 2016, 36(4): 1049-1097.
[6]	徐润章, 丁云华. GLOBAL SOLUTIONS AND FINITE TIME BLOW UP FOR DAMPED KLEIN-GORDON EQUATION[J]. 数学物理学报(英文版), 2013, 33(3): 643-652.
[7]	Nedra Moalla. A CHARACTERIZATION OF SCHECHTER´S ESSENTIAL SPECTRUM BY MEAN OF MEASURE OF NON-STRICT-SINGULARITY#br# AND APPLICATION TO MATRIX OPERATOR[J]. 数学物理学报(英文版), 2012, 32(6): 2329-2340.
[8]	Xianpeng Hu, Dehua Wang. FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY[J]. 数学物理学报(英文版), 2012, 32(1): 109-128.
[9]	陈旭忠, 沈一兵. A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS[J]. 数学物理学报(英文版), 2010, 30(4): 1053-1064.
[10]	胡耀忠, 龙红卫. ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS[J]. 数学物理学报(英文版), 2009, 29(3): 599-608.
[11]	闫宝强. MULTIPLE POSITIVE SOLUTIONS BOUNDARY VALUE PROBLEMS FOR SUPERLINEAR SECOND ORDER SINGULAR WITH DERIVATIVE DEPENDENCE[J]. 数学物理学报(英文版), 2008, 28(4): 851-864.
[12]	姚仰新; 沈尧天. ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p[J]. 数学物理学报(英文版), 2006, 26(2): 209-219.
[13]	贾高. REGULARITIES AND SINGULARITIES OF THE ENERGY MINIMIZERS OF THE HEISENBERG GROUP TARGETS[J]. 数学物理学报(英文版), 2003, 23(4): 447-.