On the bidomain problem with FitzHugh-Nagumo transport

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 <= 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 Pruss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Pruss et al in J Differ Equ 264:2028-2074, 2018), and qualitative theory of evolution equations.