  This paper aims to study the relationship between the timelike extremal
hypersurfaces and the classical minimal surfaces. This target also gives the
long time dynamics of timelike extremal hypersurfaces in Minkowski spacetime
$\mathbb{R}^{1+M}$ with the dimension $2\leq M\leq7$. In this dimension, the
stationary solution of timelike extremal hypersurface equation is the solution
of classical minimal surface equation, which only admits the hyperplane
solution by Bernstein theorem. We prove that this hyperplane solution as the
stationary solution of timelike extremal hypersurface equation is asymptotic
stablely by finding the hidden dissipative structure of linearized equation.
Here we overcome that the vector field method (based on the energy estimate and
bootstrap argument) is lose effectiveness due to the lack of time-decay of
solution for the linear perturbation equation. Meanwhile, a global well-posed
result of linear damped wave with variable time-space coefficients is
established. Hence, our result construct a unique global timelike non-small
solution near the hyperplane.
