Consider the boundary value problem:
(1) $\qquad u_{xx}  + f(u)  =  0, \quad        |x|<1,\ t>0,  $
(2) $\qquad u_t(-1,t) - u_x(-1,t) = 0, \quad    t>0           $
(3) $\qquad u_t(1,t)  +  u_x(1,t)  = 0,\quad   t>0,          $   
where $f \in C^\infty$. This problem is a simple example of a nonlinear elliptic system with a dynamic boundary condition.

Our main aim is to give an example of a nonlinearity $f$ such that the maximal existence interval of a solution  of (1)-(3) is not right open.  More precisely,  the     solution is  defined and  smooth    on $[-1,1] \times [0,t_{\max}]$, for some $t_{\max}>0$,  but it cannot  be continuously  extended   (as a solution)  to   $[-1,1] \times [0,t_{\max}+\epsilon]$ for any  $\epsilon >0$.   This means that nonexistence does  not occur  by   blow-up   of  any kind,  and  no continuation  theorem is  valid for (1)-(3),  in general.