We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on R d , equipped with power weights w(x) = |x| γ , γ > −d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.2000 Mathematics Subject Classification. 46E35, 46E40.

We investigate the properties of a class of weighted vector-valued Lp-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal Lp-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded H ∞ -calculus, interpolation theory, and operator sums.2000 Mathematics Subject Classification. 46E35, 47A60.

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.2000 Mathematics Subject Classification. Primary: 93B05. Secondary: 35K20, 93B07. Key words and phrases. Parabolic problems, dynamic boundary conditions, surface diffusion, Carleman estimate, null controllability, observability estimate.We thank the Deutsche Forschungsgemeinschaft which supported this research within the grants ME 3848/1-1 and SCHN 570/4-1. M.M. thanks L.M. for a very pleasant stay in Marrakesh, where parts of this work originated.

The mass-based Maxwell-Stefan approach to one-phase multicomponent reactive mixtures is mathematically analyzed. It is shown that the resulting quasilinear, strongly coupled reaction-diffusion system is locally well-posed in an Lp-setting and generates a local semiflow on its natural state space. Solutions regularize instantly and become strictly positive if their initial components are all nonnegative and nontrivial. For a class of reversible mass-action kinetics, the positive equilibria are identified: these are precisely the constant chemical equilibria of the system, which may form a manifold. Here the total free energy of the system is employed which serves as a Lyapunov function for the system. By the generalized principle of linearized stability, positive equilibria are proved to be normally stable.where the diffusive fluxes J k are given byNote that, by definition, k y k = 1 and k J k = 0.2000 Mathematics Subject Classification. 35R35, Secondary: 35Q30, 76D45, 76T10.

Key words Parabolic systems, inhomogeneous boundary conditions of static and relaxation type, maximal regularity, temporal weights, operator-valued functional calculus, Fourier multipliers, parabolic trace theoremWe develop a maximal regularity approach in temporally weighted Lp -spaces for vector-valued parabolic initialboundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal weights, operator-valued functional calculus, as well as localization and perturbation arguments.