ON THE HEAT FLOW OF EQUATION OF H-SURFACE

doi:10.1007/s10473-019-0516-8

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (5): 1397-1405.doi: 10.1007/s10473-019-0516-8

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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).

Abstract

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

CLC Number:

35J20

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.

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References

[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

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