Emergence of structures and advantages in cross-diffusion systems

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The main purpose of the proposed project consists in developing methods of analyzing qualitative aspects in systems of evolutionary partial differential equations modeling cross-diffusion processes. Despite a considerable increase of the mathematical literature concerned with various classes of such systems, a comprehensive understanding could so far be achieved only at levels of local or global existence theory, with questions related to qualitative behavior especially of bounded solutions widely left unanswered up to now.The particular problems to be studied, taken from application contexts especially in disciplines of biology such as spatial ecology or virology, but also in social sciences, at their mathematical core include various taxis-type or other forms of cross-diffusion, possibly coupled to degenerate diffusion or also to substantially destabilizing mechanisms in the evolution of the respective quantity determining the cross-diffusive flux. These specific settings have been chosen in such a way that their concrete mathematical structure potentially allows for a respectively deep understanding through the invention of appropriately situation-adapted techniques, but that generalizations to more general frameworks are well conceivable in subsequent steps.Beyond natural prerequisites related to topics of regularity and boundedness, a particular focus will be on the question how far the respective migration mechanisms and cross-diffusive interactions can be proved beneficial for the overall system or at least for parts thereof. This is to be pursued within suitable mathematical problem formulations appropriately quantifying phenomena of apparent relevance in the respective application context, such as dominance of accordingly motile subpopulations over static counterparts, or migration-driven support of spatial structures, or also taxis-induced persistent localization, for instance. Apart from partially addressing classical issues, e.g. of determining whether attraction-driven blow-up may occur, it is thereby particularly intended to develop new methodological approaches with potential to capture more subtle qualitative effects.By involving numerous challenges at various degrees of mathematical intricacy, but also by closely connecting the planned activities with those of the participating Chinese group, this project can moreover be expected to provide especially young researchers with considerable possibilities of professional training and collaboration. Widening perspectives and further extending the Sino-German cooperation and friendship will thus form a further vital objective of the proposed project.