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In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [6], we get a condition for the existence of smooth solutions of the Dirichlet problem for the minimal surface equation in arbitrary codimension. We also show that our condition is sharper than Wang's in [13, Theorem 1.1] provided the hyperbolic angle $\theta$ of the initial spacelike submanifold $M_{0}$ satisfies $\max_{M_{0}}{\rm cosh}\theta>\sqrt{2}$.

A NEW WAY TO DIRICHLET PROBLEMS FOR MINIMAL SURFACE SYSTEMS IN ARBITRARY DIMENSIONS AND CODIMENSIONS