In this article, we study the system with boundary degeneracy
$u_{it}-{\rm div}(a(x)|\triangledown u_{i}|^{p_{i}-2}\nabla u_i)=f_{i}(x,t,u_1,u_2),\qquad (x,t)\in\Omega_T$.
Applying the monotone iterattion technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function $a(x)$, % And if %$ \int_{\Omega} a(x)^{-\frac{1}{min{(p_1,p_2)}-1}} {\rm d}x{\rm d}t\leq C ,$ the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided $a(x)$ satisfies another restriction.