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 local assumptions on $g$, is derived by...

The existence of a periodic solution of a nonlinear equation $z^{\text{'}}+A_{0}z+B_{0}z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.