On Construction of Solutions of Hyperbolic Kirchhoff-Type Problems Involving Free Boundary and Volume Constraint

Fatima Ezahra Bentata, Ievgen Zaitsev (), Kamel Saoudi and Vladislav Kuchanskyy ()
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Fatima Ezahra Bentata: Department of Mathematics, Laboratory of Mathematics, Informatics and Systems (LAMIS), Echahid Cheikh Larbi Tebessi University-Tebessa, Tebessa 12002, Algeria
Ievgen Zaitsev: Department of Theoretical Electrical Engineering and Diagnostics of Electrical Equipment, Institute of Electrodynamics, National Academy of Sciences of Ukraine, Beresteyskiy Avenue, 56, 03057 Kyiv, Ukraine
Kamel Saoudi: College of Sciences at Dammam, University of Imam Abdulrahman Bin Faisal, Dammam 31441, Saudi Arabia
Vladislav Kuchanskyy: Department of Power-Supply Systems Optimization, Institute of Electrodynamics, National Academy of Sciences of Ukraine, Beresteyskiy Avenue, 56, 03057 Kyiv, Ukraine

Mathematics, 2025, vol. 13, issue 8, 1-24

Abstract: In this paper, we have undertaken the challenging and novel task of establishing the existence of weak solutions for four types of hyperbolic Kirchhoff-type problems: the classical hyperbolic Kirchhoff problem, the problem with a free boundary, the problem with a volume constraint, and the problem combining both a volume constraint and a free boundary. These problems are characterized by the presence of non-local terms arising from the Kirchhoff term, the free boundary, and the volume constraint, which introduces significant analytical complexities. To address these challenges, we utilize the discrete Morse flow (DMF) approach, reformulating the original continuous problem into a sequence of discrete minimization problems. This method guarantees the existence of a minimizer for the discretized functional, which subsequently serves as a weak solution to the primary problem.

Keywords: Hyperbolic Kirchhoff-type problem; volume constraint; free boundary; weak solution; discrete Morse flow (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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