On the microscopic bidomain problem with FitzHugh-Nagumo ionic transport

The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-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 $d\leq 3$, 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 combining recent results on Dirichlet-to-Neumann operators, on critical spaces for parabolic evolution equations, and qualitative theory of evolution equations.