Local Hadamard well-posedness and long-time dynamics of semilinear wave equation systems for double-wall carbon nanotubes

Abstract

In this paper, we study a class of systems of semilinear wave equations modeling double-wall carbon nanotubes. Initially, we derive the local solution employing monotone operator theory. Additionally, we also establish the uniqueness of the weak solution and its continuous dependence on the initial data. Furthermore, through the construction of a family of potential wells (which introduced in [T. Q. Minh et. al. J. Differ. Equ. 418 (2025), 374–458.]), we establish the global existence, asymptotic behavior, and blow-up of solutions for subcritical initial energy and critical initial energy, respectively. One significant advantage of our approach is the stabilization estimate, which avoids generating lower-order terms and renders the proof of asymptotic dynamics more succinct and concise. Finally, we provide evidence for the finite-time blow-up of solutions in cases with arbitrarily positive initial energy. We also prove the existence of a global attractor and provide some its properties.