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

Abstract

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.