  In this paper, we study the mathematical properties of the solution
$\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system
\begin{equation*}
  \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla
\bold{u}\right), \qquad \qquad \left(p>2\right). \end{equation*} More
precisely, we show the uniqueness and existence of solution $\bold{u}$ and
investigate a priori $L^{\infty}$ boundedness of the gradient of the solution.
Assuming that the solution decays quickly at infinity, we also prove that the
component $u^l$, $\left(1\leq l\leq k\right)$, converges to the function
$c^l\mathcal{B}$ in space as $t\to\infty$. Here, the function $\mathcal{B}$ is
the fundamental or Barenblatt solution of $p$-Laplacian equation and the
constant $c^l$ is determined by the $L^1$-mass of $u^l$. The proof is based on
the existence of entropy functional.\\ \indent As an application of the
asymptotic large time behaviour, we establish a Harnack type inequality which
makes the size of spatial average being controlled by the value of solution at
one point.
