Long-Time Dynamics of Balakrishnan–Taylor Extensible Beams

This paper is concerned with well-posedness and long-time dynamics for a class extensible beams with nonlocal Balakrishnan–Taylor and frictional damping. The related model describes vibrations in nonlinear extensible beams arising in connection with models of oscillation in pipes and supersonic panel flutter. Our main results feature the study of the nonlinear dynamical system generated by the problem. The main novelty is to explore the global Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document}-regularity (q≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 2$$\end{document}) in time of the nonlocal Balakrishnan–Taylor term and show how it generates a dissipative term that plays an important role in the asymptotic behavior of solutions, mainly in what concerns to achieve the useful property of quasi-stability in the theory of infinite-dimensional dynamical systems.