具有一般聚焦源的间接阻尼波-MGT系统的稳定与不稳定势阱动力学
Stable and Unstable Potential-Well Dynamics for an Indirectly Damped Wave-MGT System with General Focusing Sources
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- Jeonbuk National University（全北国立大学）
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中文总结 AI 辅助
研究间接阻尼波-MGT系统，通过势阱方法建立局部适定性、全局解与爆破的完整动力学分类，并给出稳定轨迹的收敛性判据。
中文摘要 AI 辅助
我们在一个有界域上研究一个保守的半线性波动方程，该方程通过零阶相互作用与一个耗散的Moore-Gibson-Thompson方程耦合。波动分量没有直接阻尼，并由一个一般的聚焦源$f(u)$驱动。增广变量$w=v+\tau v_t$揭示了精确的耦合能量和一个强制性的势阱几何。源假设通过$H_\theta(s)=\frac1\theta sf(s)-F(s)$，$F(s)=\int_0^s f(r)\\,d r$，$\theta>2$来表述。在$L^2$-次临界$C^1$增长、原点处的小性、$H_\theta$的非负性和径向单调性以及一个非平凡的聚焦条件下，我们建立了任意有限能量数据的局部适定性、精确能量恒等式和延拓交替。对于非零耦合，线性化半群是强稳定的，而波分支展开排除了一致指数稳定性和正时间紧性。在耦合阱深以下，稳定集是正不变的并产生全局解，而具有负Nehari泛函的数据在有限时间内爆破，无需对初始速度施加符号条件。在临界水平$E(0)=d$，非零耦合产生一个完整的三分法：稳定进入、有限时间爆破或平稳Nehari基态。在相同的耦合条件下，每个稳定轨迹弱收敛到零，无需任何紧性假设。一个重整化的高频恒等式表明，累积的非线性高-低通量的消失等价于轨道的相对紧性和在自然能量空间中的强收敛。我们给出了几个充分判据，包括非线性力在$L^2(\Omega)$中的全变差有限。
英文摘要
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 $H_θ(s)=\frac1θsf(s)-F(s)$, $F(s)=\int_0^s f(r)\,d r$, $θ>2$. 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 $L^2(Ω)$.
