Analysis and optimal control of quasilinear parabolic evolution equations in divergence form on rough domains

The subject of this thesis is the theory of quasilinear parabolic evolution equations in divergence form and their optimal control. The equations are posed in an abstract form in dual spaces of Sobolev spaces with partially vanishing traces. This allows to deal with mixed boundary conditions even for quite nonsmooth geometries of the underlying spatial set. For optimal control problems subject to such quasilinear equations, the possibility of blow-up of solutions poses a fundamental problem, in particular in the presence of state constraints. The main tools for the analytical results are function space theory, elliptic regularity, and maximal parabolic regularity. These tools are reviewed and established via a general abstract approach, highlighting the essential assumptions in view of the low regularity of the spatial set, and lead to existence and uniqueness of local-in-time-solutions to the quasilinear equations. A new result on uniform Hölder-norm bounds for solutions to nonautonomous evolution equations allows to infer existence and uniqueness of global-in-time solutions under additional assumptions. The optimal control problem is reduced to the set of controls admitting global-in-time solutions which allows to derive and formulate classical first-order necessary optimality conditions. The results are applied to an optimal control problem for a real-world application, the thermistor problem, and numerical experiments highlight the necessity to deal with the model as complicated as it is.