In this paper, we consider the existence and asymptotic behavior of solutions of the following problem: $u_{tt}$ -(t, x) - (∥∇u(t, x)∥(equation omitted) ＋ ∥∇v(t, x) (equation omitted)$^{gamma}$ $Delta$u(t, x)＋$delta$│ $u_{t}$ (t, x)│sup p-1/ $u_{t}$ (t, x) = $mu$│u(t, x) $^{q-1}$u(t, x), x$in$$Omega$, t$in$[0, T], $v_{tt}$ (t, x) - (∥∇uu(t, x) (equation omitted) ＋ ∥∇v(t, x) (equation omitted)sup ${gamma}$/ $Delta$v(t, x)＋$delta$ │ $v_{t}$ (t, x)│sup p-1/ $u_{t}$ (t, x) = $mu$ u(t, x) $^{q-1}$u(t, x), x$in$$Omega$, t$in$[0, T], u(0, x) = $u_{0}$ (x), $u_{t}$ (0, x) = $u_1$(x), x$in$$Omega$, u(0, x) = $v_{0}$ (x), $v_{t}$ (0, x) = $v_1$(x), x$in$$Omega$, u│∂$Omega$=v│∂$Omega$=0 T > 0, q > 1, p $geq$1, $delta$ > 0, $mu$ $in$ R, ${gamma}$ $geq$ 1 and $Delta$ is the Laplacian in $R^{N}$.X> N/.