Global Well-Posedness and Conditional Asymptotic Stability for a Coupled Wave-MGT System with Logarithmic Nonlinearity

TL;DR

This paper establishes global well-posedness and conditional asymptotic stability for a coupled wave-MGT system with a logarithmic nonlinearity, revealing energy structures and spectral obstructions to exponential decay.

Contribution

It introduces a novel coupled energy framework and constructs a potential well to prove well-posedness and stability without assuming exponential decay.

Findings

Global weak solutions exist below the well depth.

Energy dissipates strictly through the MGT component.

Spectral analysis shows high frequency obstructions to exponential decay.

Abstract

We study a coupled system formed by a conservative wave equation and a dissipative Moore-Gibson-Thompson (MGT) equation on a bounded domain. The wave component is driven by the logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$, $2<\gamma<\frac{2(n-1)}{n-2}$, and carries no direct damping. Rather than employing cross-multiplier arguments, we introduce the coupled variable $w=v+\tau v_{t}$, which reveals the exact energy structure associated with the interaction term. This formulation yields a genuine coupled energy together with a coercive quadratic form $Q_{\alpha}(u,w)=\norm{\nabla u}_{2}^{2}+\norm{\nabla w}_{2}^{2}+2\alpha(u,w)$, provided that $|\alpha|<\lambda_{1}$. Based on this structure, we construct a coupled potential well and prove global well-posedness of weak solutions for initial data lying below the corresponding well depth and inside the stable set. We also show that the energy is strictly dissipative through the MGT component. In addition, a modal analysis of the linearized system identifies a high frequency spectral obstruction to uniform exponential decay, quantifying the weakness of the dissipation transfer to the wave branch. Finally, assuming the relative compactness of the trajectory in the natural energy space and imposing $0<|\alpha|<\lambda_1$, we apply LaSalle's invariance principle to establish conditional asymptotic stability of the zero equilibrium. The result provides a structurally consistent indirect stabilization theorem for the coupled wave--MGT dynamics without relying on unjustified exponential decay claims.