# On the existence-uniqueness and computation of solution of a coupled PDE-ODE system with application to cardiac electric activity

## TL;DR

This paper investigates the existence, uniqueness, and numerical computation of solutions for a coupled PDE-ODE system modeling cardiac electrical activity, combining theoretical analysis with finite element methods and simulations.

## Contribution

It provides a rigorous proof of existence and uniqueness for a degenerate reaction-diffusion system in cardiac modeling, and develops a finite element based numerical scheme with simulations.

## Key findings

- Global existence of solutions established
- Uniqueness proven under specific nonlinear conditions
- Numerical solutions successfully computed with FreeFem++

## Abstract

In this study, we consider a system of degenerate reaction-diffusion equations, which govern the electric activity in the heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by Morris $\&$ Lecar. The global existence of a solution is established based on regularization argument using Fedo-Galerkin/Compactness approach. The uniqueness of the solution is shown based on Gronwell's Lemma upon some special treatment of nonlinear terms. The system of the continuous space-time model is first reduced to a semi-discrete time-dependent system based on finite element formulation, and then the fully discrete system is derived using the Backward Euler time stepping scheme. The numerical solution obtained using FreeFem++ are presented.