On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions
Monica De Angelis
TL;DR
The work studies a nonlinear integro-differential model linked to the perturbed sine-Gordon equation under Neumann boundary conditions, focusing on how a small diffusion term $ε$ alters wave dynamics. It develops a Fourier-series Green function, derives an ε-independent decay bound, and analyzes the remainder between the full solution and its hyperbolic limit. The paper also constructs explicit hyperbolic-sine-Gordon solutions with bounded derivatives and proves a Gronwall-based estimate showing diffusion effects vanish as $ε→0$ over long time scales. These results quantify the parabolic-to-hyperbolic transition and provide explicit solvable regimes relevant to Josephson-junction models and other Neumann-boundary dissipative systems.
Abstract
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 high order derivative with small diffusion coefficient , 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 e tends to zero.