A semigroup approach to hyperbolic systems on networks

Partial differential equations are not only purely mathematical objects but often they describe some natural phenomena. Given an equation we need to solve three basic problems. The first one is to choose an appropriate method to analyze this equation which is usually related to its type. We should also decide which space is the most reasonable to consider this equation in depending on the interpretation of unknown functions. The last but not least is to pose boundary conditions that will not only reflect the nature of the modeled phenomenon but will also ensure that the obtained initial-boundary value problem is well-posed.

The aim of this talk is to present some results concerning hyperbolic systems of PDEs on networks. We start with some examples of telegraph-type equations on a single interval and then move to more general systems on a metric graph. The main part of the presentation is focused on constructing appropriate transmission conditions in the vertices of the underlying graph. We also justify the choice of a suitable function space. These steps lead to an initial-boundary value problem that can be investigated using the apparatus of the theory of semigroups of operators.