On the microscopic bidomain problem with FitzHugh–Nagumo ionic transport

摘要

The microscopic bidomain problem with FitzHugh–Nagumo ionic transport is studied in the Lpdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L_p$$end{document}–Lqdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L_q$$end{document}-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension d≤3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$dle 3$$end{document}, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria parallel those of the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained by combining recent results on Dirichlet-to-Neumann operators (Prüss in J Integral Equ Appl, 2018; Prüss and Simonett in Moving interfaces and quasilinear parabolic evolution equations, monographs in mathematics, vol 105. Birkh?user, Basel, 2016), on critical spaces for parabolic evolution equations (Prüss et al. in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.