u_{xx}  + f(u)  =  0,                 |x|<1,    t>0,                      (1)

u_t(-1,t) - u_x(-1,t) = 0,        t>0,                                   (2)

u_t(1,t)  +  u_x(1,t)  = 0,            t>0,                             (3)

$$
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.