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

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.