The following initial-boundary value problem for a degenerate totally nonlinear parabolic equation is considered: \[ u_t=\beta (\phi (x,u_x)u_{xx} + f(x,u,u_x)),\quad (x,t)\in (0,1)\times (0,\infty ), \] \[ u_x(j,t)\in (-1)^j\beta_{j}(u(j,t)),\;j=0,1, \quad u(x,0)=u_0(x), \] as well as its higher space dimensional analogue. Here \(\beta_0\) and \(\beta_1\) are maximal monotone graphs in \(\mathbb{R}\times \mathbb{R},\) and \(\beta (t)\) or \(\beta' (t)\) might equal zero for some \(t.\) It is shown by the method of lines (constructing the Rothe function) and nonlinear semigroup theory that this problem has a unique global solution.