Abstract. The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

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We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.

The model considered consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins. We provide sufficient conditions for uniqueness of monomer-preserving weak solutions. In addition, we also prove existence of weak solutions under rather general assumptions on the involved degradation rates.

We consider the diffusive continuous coagulation-fragmentation equations with and without scattering and show that they admit unique strong solutions for a large class of initial values. If the latter values are small with respect to a suitable norm, we provide sufficient conditions for global-in-time existence in the absence of fragmentation.