Existence of solutions for a Kirchhoff hyperbolic problems in bounded domains

Abstract

This 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.