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We investigate a parabolic-elliptic problem, where the time derivative is multiplied by a coefficient which may vanish on time-dependent spatial subdomains. The linear equation is supplemented by a nonlinear Neumann boundary condition $-\partial u/\partial {\nu }_A=g(·,·,u)$ with a locally defined, $L_r$-bounded function $g(t,·,\xi )$. We prove the existence of a local weak solution to the problem by means of the Rothe method. A uniform a priori estimate for the Rothe approximations in $L_{\infty }$, which is required by the assumptions on $g$, is derived by a technique...