Non-autonomous set-valued dynamical processes: Asymptotics and applications

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The main thrust of this project concerns the long-term dynamical behaviour of non-autonomous setvalued processes on non-linear metric state spaces with different topologies when considered as the domain and the image space. These provide an abstract framework for diverse types of differential equations, difference equations and further problems occurring in modelling. Such nonunique evolution occurs naturally in, for example, control systems, and also when we are not able to establish it for Cauchy problems, such as in the 3-dimensional Navier-Stokes equations. Banach spaces with many standard topologies spaces (like the norm and weak topologies) are often adequate as state spaces, but there are important applications, e.g., involving constraints, where they are not. Improved and more widely applicable conditions sufficient for the existence of attractors are expected by a more general choices of state spaces and topologies on them when considered as the domain or the image spaces. Existence and stability of solutions have a rather introductory character, but are nevertheless essential. This project focuses on the asymptotic properties, i.e. in particular, on global and pullback attractors. The theory of attractors is extended for non-autonomous set-valued processes. Particular emphasis will be given to examples, both to illustrate and motivate the results. The evolution of shapes (i.e. possibly non-smooth sets in the Euclidean space) and their boundaries will also be investigated.