Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

The time-like extremal surface is an interesting model which may be viewed as simple but nontrivial examples of membrane in field theory. In this paper, we first consider the global weak solutions to the timelike extremal surface in Minkowski space R1+(1+n). Furthermore, we can get global smooth solutions to the timelike extremal hypersurface for initial data sufficiently close to the arbitrary time-like plane a0t+a1x1+⋯+anxn+b=0 in Minkowski space R1+(n+1), but not in the initial plane t=0.