The paper studies the Cauchy problem for a nonlinear degenerate parabolic equation. The equation with state \(u(x, t)\), \(x\in \mathbb R\), \(t> 0\) is described by \[ u_t= (u^\alpha)_{xx}+ a|u_x|^\lambda+ cv(x,t),\quad u(x,0)= f(x), \] where both \(u\) and \(f\) are nonnegative, and the parameters are such that \(ac\neq 0\), \(1<\alpha< 2\), and \(0<\lambda< 2/(3- \alpha)\). The stabilization problem is to find a suitable input \(v\) as a function of \(u\) (viewed as a kind of feedback control) such that \(\lim_{t\to\infty}u(x,t)= 0\) for all \(x\in \mathbb R\). A set of nonnegative functions \(f\) in \(C(\mathbb R)\), say \(G\), is introduced, such that \(f\) is bounded from above by \(\text{const}|x|^{\nu_0}\), \(\nu_0= (2-\lambda)/(\alpha- \lambda)\) when \(|x|\to\infty\), where the constant is determined by the above parameters and the induced pseudocharacteristic equation. The principal result is stated as follows: By setting \(v(x, t)= ru(x, t)^\beta\) with \(|r|\leq 1\), \(rc< 0\), and \(\beta= \lambda(2 -\alpha)/(2-\lambda)\), the stabilization is achieved as long as the initial state \(f\) belongs to the set \(G\).