On a resolvent estimate for bidomain operators and its applications

TL;DR

This paper establishes resolvent estimates for bidomain operators, proving existence, uniqueness, and regularity of solutions in $L^p$ spaces, with applications to electrophysiological wave propagation modeling in the heart.

Contribution

It introduces an $L^ abla$ resolvent estimate for bidomain operators, enabling the construction of strong solutions and demonstrating the generation of $C_0$-analytic semigroups in $L^p$ spaces.

Findings

Proved existence and uniqueness of strong solutions in $L^p$ spaces.

Derived an $L^ abla$ resolvent estimate for bidomain operators.

Showed bidomain operators generate $C_0$-analytic semigroups.

Abstract

We study bidomain equations that are commonly used as a model to represent the electrophysiological wave propagation in the heart. We prove existence, uniqueness and regularity of a strong solution in $L^p$ spaces. For this purpose we derive an $L^\infty$ resolvent estimate for the bidomain operator by using a contradiction argument based on a blow-up argument. Interpolating with the standard $L^2$-theory, we conclude that bidomain operators generate $C_0$-analytic semigroups in $L^p$ spaces, which leads to construct a strong solution to a bidomain equation in $L^p$ spaces.