全次临界对数非线性强阻尼波-MGT系统的全局存在性与有限时间爆破
Global existence and finite-time blow-up for a strongly damped wave--MGT system with fully subcritical logarithmic nonlinearity
- Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University（全北国立大学数学系与纯粹与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究强阻尼波-MGT系统在全次临界对数非线性下的动力学，通过增广变量和势阱结构，证明稳定集内全局指数衰减、不稳定集内有限时间爆破。
AI中文摘要：
我们研究了一个具有对数源项$f(u)=|u|^{\gamma-2}u\ln|u|$的耦合强阻尼波-摩尔-吉布森-汤普森（MGT）系统，其中指数$\gamma$处于完全Sobolev次临界范围$2<\gamma<2^*:=2n/(n-2)$。主要困难在于，在该完全范围内，对数非线性不具备较低次临界区域中可用的标准紧致性和差分估计。通过引入增广变量$w=v+\tau v_t$，我们推导了强阻尼系统的精确能量耗散恒等式，并利用了相关的耦合势阱结构。对于局部理论，采用Faedo-Galerkin方案，结合一个封闭的非线性微分不等式和空间区域分裂论证，得到了局部存在性。强阻尼提供了在整个次临界范围内证明唯一性所需的额外正则性，并建立了延拓原理。在势阱深度$d_\alpha$以下，我们获得了尖锐的动力二分性：初始数据位于稳定集中的解全局存在且指数衰减，而初始数据位于不稳定集中的解在有限时间内爆破。爆破论证揭示了增广公式特有的结构相消：一旦通过对数贡献通过精确能量恒等式重建，Levine凹性泛函中不利的MGT残差就被代数吸收。这使得凹性方法无需对初始速度施加任何额外符号条件即可闭合。
英文摘要：
We study a coupled strongly damped wave--Moore--Gibson--Thompson (MGT) system with logarithmic source $f(u)=|u|^{γ-2}u\ln|u|$ in the full Sobolev-subcritical range $2<γ<2^*:=2n/(n-2)$. The main difficulty is that, in this full range, the logarithmic nonlinearity does not admit the standard compactness and difference estimates available in the lower subcritical regime. Using the augmented variable $w=v+τv_t$, we derive the exact energy-dissipation identity for the strongly damped system and exploit the associated coupled potential-well structure. For the local theory, a Faedo--Galerkin scheme combined with a closed nonlinear differential inequality and a spatial domain-splitting argument yields local existence. The strong damping provides the additional regularity needed to prove uniqueness throughout the full subcritical range and to establish a continuation principle. Below the well depth $d_α$, we then obtain a sharp dynamical dichotomy: solutions with initial data in the stable set exist globally and decay exponentially, whereas solutions with initial data in the unstable set blow up in finite time. The blow-up argument reveals a structural cancellation specific to the augmented formulation: once the logarithmic contribution is reconstructed through the exact energy identity, the unfavorable MGT residual in Levine's concavity functional is absorbed algebraically. This allows the concavity method to close without imposing any additional sign condition on the initial velocities.
