Existence and Uniqueness of Solutions for Coupled Hybrid Systems of Differential Equations

TL;DR

This paper establishes local and global existence results for solutions to coupled systems of ODEs and PDEs, modeling interacting agents influenced by reaction-diffusion signals, generalizing previous frameworks in collective motion studies.

Contribution

It introduces new existence theorems for hybrid ODE-PDE systems with space-time dependent coefficients, extending prior models of collective cell motion and chemotaxis.

Findings

Local existence of solutions for coupled ODE-PDE systems.

Global existence under broader conditions.

Generalization of models for collective cell dynamics.

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.