GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

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GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

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doi:10.1016/S0252-9602(18)30780-X

Acta mathematica scientia,Series B ›› 2018, Vol. 38 ›› Issue (3): 745-755.doi: 10.1016/S0252-9602(18)30780-X

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Dexing KONG¹, Qi LIU²

1. School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China;
2. Department of Applied Mathematics, College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Received:2016-12-12 Revised:2017-09-20 Online:2018-06-25 Published:2018-06-25
Contact: Qi LIU E-mail:21106052@zju.edu.cn
Supported by:
This work is supported in part by the NNSF of China (11271323, 91330105), the Zhejiang Provincial Natural Science Foundation of China (LZ13A010002), and the Science Foundation in Higher Education of Henan (18A110036).

Abstract:

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation
(∂²g_ij)/(∂t²) + μ/((1 + t)^λ) (∂g_ij)/∂t=-2R_ij,
on Riemann surface. On the basis of the energy method, for 0 < λ ≤ 1, μ > λ + 1, we show that there exists a global solution g_ij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric g_ij remains uniformly bounded.

Key words: Hyperbolic geometry flow, time-dependent damping, classical solution, energy method, global existence

Dexing KONG, Qi LIU. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J].Acta mathematica scientia,Series B, 2018, 38(3): 745-755.

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