Existence and Uniqueness of Solutions for Coupled Hybrid Systems of Differential Equations (1807.03025v1)

Abstract: 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.