Of concern is the following totally nonlinear parabolic equation, as well as its higher space dimensional analogue \eqalign{u_t(x,t)& =\beta(\phi(x,u_x)u_{xx}+f(x,u,u_x)),\qquad (x,t)\in(0,1)\times(0,\infty)\cr u_x(j,t)&\in(-1)^j\beta_j(u(j,t)),\qquad j=0,1\cr u(x,0)&=u_0(x).\cr} Here β0and β1are maximal monotone graphs inR×R, and β(t) or β′(t) might equal zero for somet, at which the equation is not uniformly parabolic. It is shown by the method of lines and nonlinear operator semigroup theory that the equation has a unique global solution.