  We introduce a monotonicity-based method for studying input-to-state
stability (ISS) of nonlinear parabolic equations with boundary inputs. We first
show that a monotone control system is ISS if and only if it is ISS w.r.t.
constant inputs. Then we show by means of classical maximum principles that
nonlinear parabolic equations with boundary disturbances are monotone control
systems.
  With these two facts, we establish that ISS of the original nonlinear
parabolic PDE with constant \textit{boundary disturbances} is equivalent to ISS
of a closely related nonlinear parabolic PDE with constant \textit{distributed
disturbances} and zero boundary condition. The last problem is conceptually
much simpler and can be handled by means of various recently developed
techniques. As an application of our results, we show that the PDE backstepping
controller which stabilizes linear reaction-diffusion equations from the
boundary is robust with respect to additive actuator disturbances.
