We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation \(u_ t=\text{div} (| u |^{m- 1}| \nabla u |^{p-2} \nabla u)\) in a cylinder \(\Omega \times \mathbb{R}^ +\), with initial condition \(u(x,0) =u_ 0(x)\) in \(\Omega\) and vanishing on the parabolic boundary \(\partial \Omega \times \mathbb{R}^ +\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\), the exponents \(m\) and \(p\) satisfy \(m+p \geq 3\), \(p>1\), and the initial datum \(u_ 0\) is in \(L^ 1(\Omega)\).