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In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to \begin{document}$H^{-1}$\end{document} with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in \begin{document}$H^{-1}$\end{document} with order one.

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

We construct and investigate local invariant manifolds for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions and study their attractivity properties. In a companion paper we have developed the corresponding solution theory. Examples for the class of systems considered are reaction--diffusion systems or phase field models with dynamical boundary conditions and to the two--phase Stefan problem with surface tension.

We prove existence and uniquences of positive solutions of an age-structured population equation of McKendrick type with spatial diffusion in $L^1$. The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [19], [24], [30], and [31] to the non-autonomous situation. We use the theory of evolution semigroups and extrapolation spaces.