Archiv der Mathematik

, Volume 111, Issue 3, pp 313–327 | Cite as

On the bidomain problem with FitzHugh–Nagumo transport

  • Matthias Hieber
  • Jan Prüss


The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, 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 \(d\le 4\), 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 are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.


Bidomain operator Maximal \(L_p\)-regularity FitzHugh–Nagumo transport Critical spaces Global existence Stability 

Mathematics Subject Classification

35K50 92C35 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institut für MathematikMartin-Luther-Universität Halle-WittenbergHalleGermany

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