Initial-boundary value problem for the equation of timelike extremal surfaces in Minkowski space

This paper investigate the mixed initial-boundary value problem for the equation of timelike extremal surfaces in Minkowski space R1+(1+n) in the first quadrant. Under the assumptions that the initial data are bounded and the boundary data are small, we prove the global existence and uniqueness of the C2 solutions of the initial-boundary value problem for this kind of equation. Based on the existence results on global classical solutions, we also show that, as t tends to infinity, the first order derivatives of the solutions approach C1 traveling wave, under the appropriate conditions on the initial and boundary data. Geometrically, this means the extremal surface approaches a generalized cylinder which is an exact solution.

[1]  D. Kong,et al.  Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields , 2007 .

[2]  Yi Zhou,et al.  Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems , 2007 .

[3]  Qiang Zhang,et al.  The Dynamics of Relativistic Strings Moving in the Minkowski Space $$\mathbb{R}^{1+n}$$ , 2006 .

[4]  W. Dai.  Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy* , 2006 .

[5]  Yi Zhou,et al.  The equation for time-like extremal surfaces in Minkowski space R2+n , 2006 .

[6]  Tatsien Li,et al.  Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems , 2004 .

[7]  D. Chae,et al.  Global existence for small initial data in the Born–Infeld equations , 2003 .

[8]  Y. Brenier.  Some Geometric PDEs Related to Hydrodynamics and Electrodynamics , 2003, math/0305009.

[9]  Tong Yang,et al.  Asymptotic Behavior of Global Classical Solutions of Quasilinear Hyperbolic Systems , 2003 .

[10]  Hans Lindblad,et al.  A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time , 2002, math/0210056.

[11]  G. Gibbons.  Born-Infeld particles and Dirichlet p-branes , 1997, hep-th/9709027.

[12]  B. Barbashov.  Integrals of periodic motion for the classical equations of a relativistic string with masses at its ends , 1997 .

[13]  Yi Zhou,et al.  Global classical solutions for general quasilinear hyperbolic systems with decay initial data , 1997 .

[14]  J. Hoppe,et al.  Some classical solutions of relativistic membrane equations in 4 space-time dimensions , 1994, hep-th/9402112.

[15]  R. Kohn,et al.  The initial‐value problem for measure‐valued solutions of a canonical 2 × 2 system with linearly degenerate fields , 1991 .

[16]  A. M. Chervyakov,et al.  General solutions of nonlinear equations in the geometric theory of the relativistic string , 1982 .

[17]  A. Chervjakov,et al.  Infinite relativistic string with a point-like mass , 1978 .

[18]  Shing-Tung Yau,et al.  Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces , 1976 .

[19]  Yi Zhou.  GLOBAL CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY , 2004 .

[20]  Zhou Yi,et al.  Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems , 1994 .

[21]  C. Gu.  COMPLETE EXTREMAL SURFACES OF MIXED TYPE IN 3-DIMENSIONAL MINKOWSKI SPACE , 1994 .

[22]  Gui-Qiang G. Chen,et al.  Propagation and cancellation of oscillations for hyperbolic systems of conservation laws , 1991 .

[23]  C. Gu.  Extremal Surfaces of Mixed Type in Minkowski Space R n+1 , 1990 .

[24]  A. Bressan.  Contractive metrics for nonlinear hyperbolic systems , 1988 .