Bentata, Fatima Ezahra2026-01-212026-01-212025-12-11http//localhost:8080/jspui/handle/123456789/13780This study is dedicated to investigating the existence of weak solutions for hyperbolic Kirchhoff-type problems, considering cases both with and without volume constraints and free boundaries. We employ the hyperbolic discrete Morse flow, which transforms the original problem into a sequence of mini- mization problems defined at discrete time intervals. This process ensures the existence of a minimizer for the discretized functional, which corresponds to the solution of the discretized problem and subsequently provides a weak so- lution to the original problem. The non-local terms arising from the Kirchhoff component, volume constraint, and free boundary condition present significant challenges that require innovative methods to tackle. These complexities are a key focus of our research. Furthermore, we present numerical simulations to better illustrate our results’ physical implications.enHyperbolic Kirchhoff type problem, Free boundary, Volume constraint, Weak solution, Discrete Morse flow, Numerical simulationThesis