Abstract
In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal {{L}}^p-regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo,Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.
Highlights
Given a sectorial operator A on a Banach space X, the problem of maximal regularity for the inhomogenous Cauchy problem u (t) + Au(t) = f (t), 0 < t < T u(0) = 0 (ACP)received a lot of attention since the pioneering articles by Da Prato–Grisvard [10], Dore–Venni [13] and Lunardi [22]
The results are applied to the bidomain equations with FitzHugh–Nagumo transport
As counterpart to the Arendt–Bu theorem characterizing maximal periodic Lp-regularity for 1 < p < ∞ and X being a UMD-space, we provide a short derivation of a periodic version of Theorem 1.1
Summary
As the classical Da Prato–Grisvard theorem, it holds for 1 ≤ p < ∞ and arbitrary Banach spaces X To formulate this result, we define the periodicity of a measurable function as follows. Our approach to the existence of unique, strong T-periodic solutions to the bidomain equations in a neighborhood of stable equilibrium points is the semilinear version of the periodic Da Prato–Grisvard theorem in Sect. A very different approach to periodic solutions to the bidomain equations was developed by Giga, Kajiwara, and Kress [16] They showed the existence of a strong, periodic solution to the bidomain equations for arbitrary large f ∈ L2( ) based on a weak-strong uniqueness argument. 4, we present our second main result, the existence of a unique, strong T-periodic solution to the bidomain equations subject to a large class of models for the ionic transport.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
