Existence and uniqueness of solutions of a class of 3rd order dissipative problems with various boundary conditions describing the Josephson effect

TL;DR

This paper establishes the existence and uniqueness of solutions for a broad class of third-order dissipative problems, including models relevant to superconductivity and viscoelastic materials, under various boundary conditions.

Contribution

It provides a rigorous mathematical proof of solution existence and uniqueness for a class of quasi-linear third-order equations with diverse boundary conditions, encompassing models like the Josephson effect.

Findings

Proves existence of solutions for the class of equations.

Establishes uniqueness of solutions under given conditions.

Includes applications to superconductor and viscoelastic models.

Abstract

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, such as a well-known modified sine-Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.