FLARe falls into the fields of fractional nonlinear spatio-temporal evolution equations, their dynamical analysis and bifurcation structure, from a theoretical and numerical point of view. The fractional Laplacian well describes nonlocal super diffusion phenomena which naturally appear in many applications in physics, biology and economy, characterized by long-range interactions. In the theory of nonlocal operators it constitutes the counterpart of the standard Laplacian in the classical theory of partial differential equations. Despite their differences, most of the relevant questions associated to the Laplacian have an equivalent for the fractional Laplacian. The novel challenge of FLARe is to also extend the analytical and numerical tools coming from a dynamical system approach for standard diffusion equations, to treat fractional diffusion equations (e.g. amplitude equations, continuation softwares etc.). This will be done thanks to an innovative interaction between the theoretical results on fractional operators, established theory on PDEs dynamics, stability and bifurcations, new and efficient discretization techniques and the continuation software pde2path. Taken together, they are going to constitute a solid methodology to investigate the dynamical properties of nonlinear fractional diffusion equations. These aspects also constitute the long term impact of the action: a well established link of three research fields, namely fractional diffusion theory, the dynamical system approach to PDEs and continuation software available for standard diffusion equation, which are not so well connected at the moment. The new link between different communities, the important results achieved during the action, the new capabilities of the continuation software, the supervisor expertise and clear guidance, the training and the outstanding research environment provided by the host institution are going to be the springboard for the researcher to a future career in academia.
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