Existence, uniqueness and stability for a class of third order dissipative problems depending on time

TL;DR

This paper establishes existence, uniqueness, and stability results for a class of third-order dissipative PDEs with time-dependent coefficients, relevant in superconductor and viscoelastic material theories.

Contribution

It introduces a Lyapunov functional approach tailored to quasi-linear third-order equations with time-dependent coefficients, providing new theoretical insights.

Findings

Proves existence and uniqueness of solutions.

Demonstrates stability and attractivity under certain conditions.

Applies to equations in superconductor and viscoelastic material models.

Abstract

We prove new results regarding the existence, uniqueness, (eventual) boundedness, (total) stability and attractivity of the solutions of a class of initial-boundary-value problems characterized by a quasi-linear third order equation which may contain time-dependent coefficients. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials. In the proof we use a Liapunov functional V depending on two parameters, which we adapt to the characteristics of the problem.