The one-dimensional wave equation with general boundary conditions

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions αju′(j) + βju(j) + γ1–ju(1 − j) = 0, (j = 0, 1) generates a cosine family on Lp(0, 1) for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p\,{\in}\,[1,\infty)}$$\end{document}. Here αj, βj and γj are complex numbers satisfying α0, α1 ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.