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| In this paper we investigate the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ in the first quadrant. Under the assumptions that the initial data is bounded and the boundary data is small, we prove the global existence and uniqueness of the $C^{2}$ 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 $ C^{1}$ travelling wave, under the appropriate conditions on the initial and boundary datum. Geometrically, this means the extremal surface approaches a generalized cylinder. |
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| Keywords:Applied mathematics; Minkowski space; Timelike extremal surfaces; Initial-boundary value problem |
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