Stability properties for some non-autonomous dissipative phenomena proved by families of Liapunov functionals

TL;DR

This paper establishes new stability and boundedness results for a class of quasi-linear third order PDEs with time-dependent coefficients, using parameterized Lyapunov functionals, applicable in superconductor and viscoelastic material theories.

Contribution

It introduces a novel approach employing families of Lyapunov functionals tailored to the neighborhood size of the null solution for stability analysis.

Findings

Proves boundedness and stability of solutions.

Develops Lyapunov functionals adaptable to solution neighborhoods.

Applicable to equations in superconductor and viscoelastic theories.

Abstract

We prove some new results regarding the boundedness, 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 family of Liapunov functionals W depending on two parameters, which we adapt to the `error', i.e. to the size of the chosen neighbourhood of the null solution.