  In this paper we propose local and global existence results for the solution
of systems characterized by the coupling of ODEs and PDEs. The coexistence of
distinct mathematical formalisms represents the main feature of hybrid
approaches, in which the dynamics of interacting agents are driven by
second-order ODEs, while reaction-diffusion equations are used to model the
time evolution of a signal influencing them. We first present an existence
result of the solution, locally in time. In particular, we generalize the
framework of recent works presented in the literature, concerning collective
motions of cells due to mechanical forces and chemotaxis, taking into account a
uniformly parabolic operator with space-and-time dependent coefficients, and a
more general structure for the equations of motion. Then, the previous result
is extended in order to obtain a global solution.
