Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect

We prove existence and uniqueness of solutions of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in superconductor theory, in the form of various modified sine-Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.