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.