formulaThe interest for the sine-Gordon equation is explained by the various applications in differential geometry and engineering, including junctions between two superconductors, the motion of rigid pendular attached to a stretched wire, dislocations in crystals, nonlinear optics. Using similar specific techniques, the functional framework of the Sobolev space H1 0 (­) and the properties of the strongly elliptic operator, we investigate the behavior of solutions u"± to the problem (P"±) in two different cases: (i) " ! 0 and ± ¸ ±0 > 0, relative to the solutions to the following unperturbed system:formulaThe problem (P"±) is the abstract model of singularly perturbed problems of hyperbolic-parabolic type in the case (i) and of the hyperbolic-parabolic-elliptic type in the case (ii).