Existence of solutions for a hyperbolic Kirchhoff-type problem with a volume constraint

Abstract

In this study, we investigate the existence of weak solutions for a hyperbolic Kirchhoff-type problem constrained by volume. Our research is motivated by the complex nature of such problems, which involve non-local terms arising from the Kirchhoff term and the volume constraint. These non-local terms introduce significant challenges, complicating the application of traditional analytical methods and necessitating the development of innovative approaches for resolution. To address these challenges, we leverage the hyperbolic discrete Morse flow, a powerful framework that allows us to adapt traditional techniques to the unique characteristics of our problem. This framework is particularly suited for handling the intricacies of non-local terms and offers a structured way to examine the interplay between these terms and the volume constraint. Our approach includes a rigorous mathematical analysis combined with the development of novel techniques specifically designed to manage the complex interactions within the system. Through this comprehensive methodology, we aim to establish the necessary conditions under which weak solutions can exist for the given hyperbolic Kirchhoff-type problem. The findings from this study provide new insights into the behavior of such systems, potentially contributing to advancements in the broader field of partial differential equations and applied mathematics.