Qualitative properties for a class of non-autonomous semi-linear 3rd order PDE arising in dissipative problems

TL;DR

This paper investigates the stability and attractivity of solutions to a broad class of semi-linear third-order PDEs with time-dependent coefficients, using Lyapunov functionals tailored to the error term.

Contribution

It introduces a new approach with parameter-dependent Lyapunov functionals to analyze stability of non-autonomous semi-linear third-order PDEs.

Findings

Enhanced stability criteria for the PDE class.

Lyapunov functionals effectively handle time-dependent coefficients.

Improved understanding of solution behavior in dissipative systems.

Abstract

We improve results regarding the stability and attractivity of solutions $u$ of a large class of initial-boundary-value problems characterized by a semi-linear third order equation which may contain time-dependent coefficients. In the proof we use Liapunov functionals $W$ depending on two parameters, which we adapt to the 'error' $\sigma$.