Well-posedness and symmetries of strongly coupled network equations
Stefano Cardanobile, Delio Mugnolo, Robin Nittka · Journal of Physics A Mathematical and Theoretical · 2008
We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i.e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i.e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schrödinger equations on a quantum graph are discussed.