Stable and Unstable Potential-Well Dynamics for an Indirectly Damped Wave-MGT System with General Focusing Sources
Abstract
We study a conservative semilinear wave equation coupled through a zero-order interaction to a dissipative Moore--Gibson--Thompson equation on a bounded domain. The wave component carries no direct damping and is driven by a general focusing source f(u). The augmented variable w=v+τvt reveals an exact coupled energy and a coercive potential-well geometry. The source assumptions are formulated through Hθ(s)=1/θsf(s)−F(s), F(s)=∫0s f(r) d r, θ>2. Under L2-subcritical C1 growth, smallness at the origin, nonnegativity and radial monotonicity of Hθ, and a nontrivial focusing condition, we establish local well-posedness, the exact energy identity, and a continuation alternative for arbitrary finite-energy data. For nonzero coupling, the linearized semigroup is strongly stable, whereas a wave-branch expansion precludes uniform exponential stability and positive-time compactness. Below the coupled well depth, the stable set is positively invariant and generates global solutions, while data with negative Nehari functional blow up in finite time without a sign condition on the initial velocities. At the critical level E(0)=d, nonzero coupling yields a complete trichotomy into stable entry, finite-time blow-up, or a stationary Nehari ground state. Under the same coupling condition, every stable trajectory converges weakly to zero without any compactness hypothesis. A renormalized high-frequency identity shows that vanishing of the accumulated nonlinear high--low flux is equivalent to relative compactness of the orbit and to strong convergence in the natural energy space. We give several sufficient criteria, including finite total variation of the nonlinear force in L2(Ω).
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BibTeX
@misc{ha2026stable,
title = {{Stable and Unstable Potential-Well Dynamics for an Indirectly Damped Wave-MGT System with General Focusing Sources}},
author = {Tae Gab Ha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13708},
eprint = {2609.13708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13708},
}
RIS
TY - UNPB
TI - Stable and Unstable Potential-Well Dynamics for an Indirectly Damped Wave-MGT System with General Focusing Sources
AU - Tae Gab Ha
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13708
UR - https://arxiv.org/abs/2609.13708
AB - We study a conservative semilinear wave equation coupled through a zero-order interaction to a dissipative Moore--Gibson--Thompson equation on a bounded domain. The wave component carries no direct damping and is driven by a general focusing source f(u). The augmented variable w=v+τv_t reveals an exact coupled energy and a coercive potential-well geometry. The source assumptions are formulated through H_θ(s)=1/θsf(s)−F(s), F(s)=∫_0^s f(r) d r, θ>2. Under L^2-subcritical C^1 growth, smallness at the origin, nonnegativity and radial monotonicity of H_θ, and a nontrivial focusing condition, we establish local well-posedness, the exact energy identity, and a continuation alternative for arbitrary finite-energy data. For nonzero coupling, the linearized semigroup is strongly stable, whereas a wave-branch expansion precludes uniform exponential stability and positive-time compactness. Below the coupled well depth, the stable set is positively invariant and generates global solutions, while data with negative Nehari functional blow up in finite time without a sign condition on the initial velocities. At the critical level E(0)=d, nonzero coupling yields a complete trichotomy into stable entry, finite-time blow-up, or a stationary Nehari ground state. Under the same coupling condition, every stable trajectory converges weakly to zero without any compactness hypothesis. A renormalized high-frequency identity shows that vanishing of the accumulated nonlinear high--low flux is equivalent to relative compactness of the orbit and to strong convergence in the natural energy space. We give several sufficient criteria, including finite total variation of the nonlinear force in L^2(Ω).
ER -