The authors consider degenerate parabolic problems of the form \[ \begin{aligned} u_t-\varphi \bigl(u(x,t) \bigr)_{xx}= f\bigl(u(x,t) \bigr),\quad & x\in (-L,L),\;t>0,\\ u(-L,t)= u(L,t)=0, \quad & t>0,\\ u(x,0)= u_0(t)\geq 0,\quad & x\in (-L,L), \end{aligned} \] where \(\varphi\) and \(f\) are sufficiently regular functions satisfying \(\varphi(0)=0\), \(\varphi'(z)>0\) for \(z>0\), and \(f(0)\geq 0\). They prove that any bounded solution converges to a single equilibrium. To a large extent the proof depends on zero-number arguments suitably adjusted to degenerate equations.