On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions

An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine-Gordon equation that in superconductivity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a higher-order derivative with small diffusion coefficient ε,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,$$\end{document} is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document} tends to zero.